Theorem rdgeqoa 32394

Description: If a recursive function with an initial value 𝐴 at step 𝑁 is equal to itself with an initial value 𝐵 at step 𝑀, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.)

Assertion

Ref	Expression
rdgeqoa	⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋))))

Proof of Theorem rdgeqoa

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1056	. 2 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → 𝑋 ∈ ω)
2		eleq1 2676	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ ω ↔ 𝑋 ∈ ω))
3	2	3anbi3d 1397	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω)))
4		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑁 +𝑜 𝑥) = (𝑁 +𝑜 𝑋))
5	4	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = 𝑋 → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)))
6		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀 +𝑜 𝑥) = (𝑀 +𝑜 𝑋))
7	6	fveq2d 6107	. . . . . 6 ⊢ (𝑥 = 𝑋 → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋)))
8	5, 7	eqeq12d 2625	. . . . 5 ⊢ (𝑥 = 𝑋 → ((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋))))
9	8	imbi2d 329	. . . 4 ⊢ (𝑥 = 𝑋 → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋)))))
10	3, 9	imbi12d 333	. . 3 ⊢ (𝑥 = 𝑋 → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋))))))
11		peano1 6977	. . . . 5 ⊢ ∅ ∈ ω
12		oa0 7483	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ On → (𝑁 +𝑜 ∅) = 𝑁)
13	12	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑁 ∈ On → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐴)‘𝑁))
14	13	eqcomd 2616	. . . . . . . . . 10 ⊢ (𝑁 ∈ On → (rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)))
15		oa0 7483	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ On → (𝑀 +𝑜 ∅) = 𝑀)
16	15	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑀 ∈ On → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘𝑀))
17	16	eqcomd 2616	. . . . . . . . . 10 ⊢ (𝑀 ∈ On → (rec(𝐹, 𝐵)‘𝑀) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅)))
18	14, 17	eqeqan12d 2626	. . . . . . . . 9 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅))))
19	18	biimpd 218	. . . . . . . 8 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅))))
20		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 ∈ ω ↔ ∅ ∈ ω))
21	20	3anbi3d 1397	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω)))
22	11	biantru 525	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ On ↔ (𝑀 ∈ On ∧ ∅ ∈ ω))
23	22	anbi2i 726	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
24		3anass 1035	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
25	23, 24	bitr4i 266	. . . . . . . . . 10 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω))
26	21, 25	syl6bbr 277	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On)))
27		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑁 +𝑜 𝑥) = (𝑁 +𝑜 ∅))
28	27	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)))
29		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑀 +𝑜 𝑥) = (𝑀 +𝑜 ∅))
30	29	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅)))
31	28, 30	eqeq12d 2625	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅))))
32	31	imbi2d 329	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅)))))
33	26, 32	imbi12d 333	. . . . . . . 8 ⊢ (𝑥 = ∅ → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 ∅))))))
34	19, 33	mpbiri 247	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
35	34	ax-gen 1713	. . . . . 6 ⊢ ∀𝑥(𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
36		sbc6g 3428	. . . . . 6 ⊢ (∅ ∈ ω → ([∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ∀𝑥(𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))))
37	35, 36	mpbiri 247	. . . . 5 ⊢ (∅ ∈ ω → [∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
38	11, 37	ax-mp 5	. . . 4 ⊢ [∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
39		peano2b 6973	. . . . 5 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
40	39	3anbi3i 1248	. . . . . . . 8 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω))
41	40	imbi1i 338	. . . . . . 7 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
42		nnon 6963	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
43		oacl 7502	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ On ∧ 𝑥 ∈ On) → (𝑁 +𝑜 𝑥) ∈ On)
44		oacl 7502	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ On ∧ 𝑥 ∈ On) → (𝑀 +𝑜 𝑥) ∈ On)
45	43, 44	anim12i 588	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑀 ∈ On ∧ 𝑥 ∈ On)) → ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On))
46	45	3impdir 1374	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On) → ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On))
47		rdgsuc 7407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 +𝑜 𝑥) ∈ On → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥))))
48		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ ((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) → (𝐹‘(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥))) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
49	47, 48	sylan9eqr 2666	. . . . . . . . . . . . . . . . 17 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ (𝑁 +𝑜 𝑥) ∈ On) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
50	49	adantrr 749	. . . . . . . . . . . . . . . 16 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
51		rdgsuc 7407	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 +𝑜 𝑥) ∈ On → (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
52	51	ad2antll 761	. . . . . . . . . . . . . . . 16 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On)) → (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
53	50, 52	eqtr4d 2647	. . . . . . . . . . . . . . 15 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ ((𝑁 +𝑜 𝑥) ∈ On ∧ (𝑀 +𝑜 𝑥) ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
54	46, 53	sylan2 490	. . . . . . . . . . . . . 14 ⊢ (((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ∧ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
55	54	ancoms 468	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
56	42, 55	syl3anl3 1368	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
57		onasuc 7495	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (𝑁 +𝑜 suc 𝑥) = suc (𝑁 +𝑜 𝑥))
58	57	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)))
59	58	3adant2 1073	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)))
60	59	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +𝑜 𝑥)))
61		onasuc 7495	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (𝑀 +𝑜 suc 𝑥) = suc (𝑀 +𝑜 𝑥))
62	61	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
63	62	3adant1 1072	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
64	63	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +𝑜 𝑥)))
65	56, 60, 64	3eqtr4d 2654	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))
66	65	ex 449	. . . . . . . . . 10 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))
67	66	imim2d 55	. . . . . . . . 9 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
68	40, 67	sylbir 224	. . . . . . . 8 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
69	68	a2i 14	. . . . . . 7 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
70	41, 69	sylbi 206	. . . . . 6 ⊢ (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
71		sbcimg 3444	. . . . . . 7 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → [suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))))
72		sbc3an 3461	. . . . . . . . 9 ⊢ ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ ([suc 𝑥 / 𝑥]𝑁 ∈ On ∧ [suc 𝑥 / 𝑥]𝑀 ∈ On ∧ [suc 𝑥 / 𝑥]𝑥 ∈ ω))
73		sbcg 3470	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑁 ∈ On ↔ 𝑁 ∈ On))
74		sbcg 3470	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑀 ∈ On ↔ 𝑀 ∈ On))
75		sbcel1v 3462	. . . . . . . . . . 11 ⊢ ([suc 𝑥 / 𝑥]𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
76	75	a1i 11	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑥 ∈ ω ↔ suc 𝑥 ∈ ω))
77	73, 74, 76	3anbi123d 1391	. . . . . . . . 9 ⊢ (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥]𝑁 ∈ On ∧ [suc 𝑥 / 𝑥]𝑀 ∈ On ∧ [suc 𝑥 / 𝑥]𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω)))
78	72, 77	syl5bb 271	. . . . . . . 8 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω)))
79		sbcimg 3444	. . . . . . . . 9 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → [suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
80		sbcg 3470	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) ↔ (rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀)))
81		sbceqg 3936	. . . . . . . . . . 11 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))
82		csbfv12 6141	. . . . . . . . . . . . 13 ⊢ ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐴)‘⦋suc 𝑥 / 𝑥⦌(𝑁 +𝑜 𝑥))
83		csbconstg 3512	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐴) = rec(𝐹, 𝐴))
84		csbov123 6585	. . . . . . . . . . . . . . 15 ⊢ ⦋suc 𝑥 / 𝑥⦌(𝑁 +𝑜 𝑥) = (⦋suc 𝑥 / 𝑥⦌𝑁⦋suc 𝑥 / 𝑥⦌ +𝑜 ⦋suc 𝑥 / 𝑥⦌𝑥)
85		csbconstg 3512	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌ +𝑜 = +𝑜 )
86		csbconstg 3512	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌𝑁 = 𝑁)
87		csbvarg 3955	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌𝑥 = suc 𝑥)
88	85, 86, 87	oveq123d 6570	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌𝑁⦋suc 𝑥 / 𝑥⦌ +𝑜 ⦋suc 𝑥 / 𝑥⦌𝑥) = (𝑁 +𝑜 suc 𝑥))
89	84, 88	syl5eq 2656	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(𝑁 +𝑜 𝑥) = (𝑁 +𝑜 suc 𝑥))
90	83, 89	fveq12d 6109	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐴)‘⦋suc 𝑥 / 𝑥⦌(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)))
91	82, 90	syl5eq 2656	. . . . . . . . . . . 12 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)))
92		csbfv12 6141	. . . . . . . . . . . . 13 ⊢ ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) = (⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐵)‘⦋suc 𝑥 / 𝑥⦌(𝑀 +𝑜 𝑥))
93		csbconstg 3512	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐵) = rec(𝐹, 𝐵))
94		csbov123 6585	. . . . . . . . . . . . . . 15 ⊢ ⦋suc 𝑥 / 𝑥⦌(𝑀 +𝑜 𝑥) = (⦋suc 𝑥 / 𝑥⦌𝑀⦋suc 𝑥 / 𝑥⦌ +𝑜 ⦋suc 𝑥 / 𝑥⦌𝑥)
95		csbconstg 3512	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌𝑀 = 𝑀)
96	85, 95, 87	oveq123d 6570	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌𝑀⦋suc 𝑥 / 𝑥⦌ +𝑜 ⦋suc 𝑥 / 𝑥⦌𝑥) = (𝑀 +𝑜 suc 𝑥))
97	94, 96	syl5eq 2656	. . . . . . . . . . . . . 14 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(𝑀 +𝑜 𝑥) = (𝑀 +𝑜 suc 𝑥))
98	93, 97	fveq12d 6109	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌rec(𝐹, 𝐵)‘⦋suc 𝑥 / 𝑥⦌(𝑀 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))
99	92, 98	syl5eq 2656	. . . . . . . . . . . 12 ⊢ (suc 𝑥 ∈ ω → ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))
100	91, 99	eqeq12d 2625	. . . . . . . . . . 11 ⊢ (suc 𝑥 ∈ ω → (⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = ⦋suc 𝑥 / 𝑥⦌(rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))
101	81, 100	bitrd 267	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))
102	80, 101	imbi12d 333	. . . . . . . . 9 ⊢ (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → [suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
103	79, 102	bitrd 267	. . . . . . . 8 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥)))))
104	78, 103	imbi12d 333	. . . . . . 7 ⊢ (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → [suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))))
105	71, 104	bitrd 267	. . . . . 6 ⊢ (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 suc 𝑥))))))
106	70, 105	syl5ibr 235	. . . . 5 ⊢ (suc 𝑥 ∈ ω → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) → [suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))))
107	39, 106	sylbi 206	. . . 4 ⊢ (𝑥 ∈ ω → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))) → [suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥))))))
108	38, 107	findes 6988	. . 3 ⊢ (𝑥 ∈ ω → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑥)))))
109	10, 108	vtoclga 3245	. 2 ⊢ (𝑋 ∈ ω → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋)))))
110	1, 109	mpcom 37	1 ⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +𝑜 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +𝑜 𝑋))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  [wsbc 3402  ⦋csb 3499  ∅c0 3874  Oncon0 5640  suc csuc 5642  ‘cfv 5804  (class class class)co 6549  ωcom 6957  reccrdg 7392   +_𝑜 coa 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451

This theorem is referenced by:  finxpreclem4  32407