Theorem rdgprc 30944

Description: The value of the recursive definition generator when 𝐼 is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
rdgprc	⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))

Proof of Theorem rdgprc

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6103	. . . . . . 7 ⊢ (𝑧 = ∅ → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘∅))
2		fveq2 6103	. . . . . . 7 ⊢ (𝑧 = ∅ → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘∅))
3	1, 2	eqeq12d 2625	. . . . . 6 ⊢ (𝑧 = ∅ → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅)))
4	3	imbi2d 329	. . . . 5 ⊢ (𝑧 = ∅ → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))))
5		fveq2 6103	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑦))
6		fveq2 6103	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑦))
7	5, 6	eqeq12d 2625	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
8	7	imbi2d 329	. . . . 5 ⊢ (𝑧 = 𝑦 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦))))
9		fveq2 6103	. . . . . . 7 ⊢ (𝑧 = suc 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘suc 𝑦))
10		fveq2 6103	. . . . . . 7 ⊢ (𝑧 = suc 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘suc 𝑦))
11	9, 10	eqeq12d 2625	. . . . . 6 ⊢ (𝑧 = suc 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦)))
12	11	imbi2d 329	. . . . 5 ⊢ (𝑧 = suc 𝑦 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦))))
13		fveq2 6103	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑥))
14		fveq2 6103	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑥))
15	13, 14	eqeq12d 2625	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
16	15	imbi2d 329	. . . . 5 ⊢ (𝑧 = 𝑥 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))))
17		rdgprc0 30943	. . . . . 6 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
18		0ex 4718	. . . . . . 7 ⊢ ∅ ∈ V
19	18	rdg0 7404	. . . . . 6 ⊢ (rec(𝐹, ∅)‘∅) = ∅
20	17, 19	syl6eqr 2662	. . . . 5 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))
21		fveq2 6103	. . . . . . 7 ⊢ ((rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
22		rdgsuc 7407	. . . . . . . 8 ⊢ (𝑦 ∈ On → (rec(𝐹, 𝐼)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
23		rdgsuc 7407	. . . . . . . 8 ⊢ (𝑦 ∈ On → (rec(𝐹, ∅)‘suc 𝑦) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
24	22, 23	eqeq12d 2625	. . . . . . 7 ⊢ (𝑦 ∈ On → ((rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦) ↔ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦))))
25	21, 24	syl5ibr 235	. . . . . 6 ⊢ (𝑦 ∈ On → ((rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦)))
26	25	imim2d 55	. . . . 5 ⊢ (𝑦 ∈ On → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦))))
27		r19.21v 2943	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑧 (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) ↔ (¬ 𝐼 ∈ V → ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
28		limord 5701	. . . . . . . . 9 ⊢ (Lim 𝑧 → Ord 𝑧)
29		ordsson 6881	. . . . . . . . 9 ⊢ (Ord 𝑧 → 𝑧 ⊆ On)
30		rdgfnon 7401	. . . . . . . . . 10 ⊢ rec(𝐹, 𝐼) Fn On
31		rdgfnon 7401	. . . . . . . . . 10 ⊢ rec(𝐹, ∅) Fn On
32		fvreseq 6227	. . . . . . . . . 10 ⊢ (((rec(𝐹, 𝐼) Fn On ∧ rec(𝐹, ∅) Fn On) ∧ 𝑧 ⊆ On) → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) ↔ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
33	30, 31, 32	mpanl12 714	. . . . . . . . 9 ⊢ (𝑧 ⊆ On → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) ↔ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
34	28, 29, 33	3syl 18	. . . . . . . 8 ⊢ (Lim 𝑧 → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) ↔ ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
35		rneq 5272	. . . . . . . . . . 11 ⊢ ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → ran (rec(𝐹, 𝐼) ↾ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧))
36		df-ima 5051	. . . . . . . . . . 11 ⊢ (rec(𝐹, 𝐼) “ 𝑧) = ran (rec(𝐹, 𝐼) ↾ 𝑧)
37		df-ima 5051	. . . . . . . . . . 11 ⊢ (rec(𝐹, ∅) “ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧)
38	35, 36, 37	3eqtr4g 2669	. . . . . . . . . 10 ⊢ ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧))
39	38	unieqd 4382	. . . . . . . . 9 ⊢ ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → ∪ (rec(𝐹, 𝐼) “ 𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧))
40		vex 3176	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
41		rdglim 7409	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = ∪ (rec(𝐹, 𝐼) “ 𝑧))
42		rdglim 7409	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, ∅)‘𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧))
43	41, 42	eqeq12d 2625	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ Lim 𝑧) → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ ∪ (rec(𝐹, 𝐼) “ 𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧)))
44	40, 43	mpan 702	. . . . . . . . 9 ⊢ (Lim 𝑧 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ ∪ (rec(𝐹, 𝐼) “ 𝑧) = ∪ (rec(𝐹, ∅) “ 𝑧)))
45	39, 44	syl5ibr 235	. . . . . . . 8 ⊢ (Lim 𝑧 → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)))
46	34, 45	sylbird 249	. . . . . . 7 ⊢ (Lim 𝑧 → (∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)))
47	46	imim2d 55	. . . . . 6 ⊢ (Lim 𝑧 → ((¬ 𝐼 ∈ V → ∀𝑦 ∈ 𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧))))
48	27, 47	syl5bi 231	. . . . 5 ⊢ (Lim 𝑧 → (∀𝑦 ∈ 𝑧 (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧))))
49	4, 8, 12, 16, 20, 26, 48	tfinds 6951	. . . 4 ⊢ (𝑥 ∈ On → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
50	49	com12 32	. . 3 ⊢ (¬ 𝐼 ∈ V → (𝑥 ∈ On → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
51	50	ralrimiv 2948	. 2 ⊢ (¬ 𝐼 ∈ V → ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
52		eqfnfv 6219	. . 3 ⊢ ((rec(𝐹, 𝐼) Fn On ∧ rec(𝐹, ∅) Fn On) → (rec(𝐹, 𝐼) = rec(𝐹, ∅) ↔ ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
53	30, 31, 52	mp2an 704	. 2 ⊢ (rec(𝐹, 𝐼) = rec(𝐹, ∅) ↔ ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
54	51, 53	sylibr 223	1 ⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ran crn 5039   ↾ cres 5040   “ cima 5041  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  reccrdg 7392

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-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-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393

This theorem is referenced by:  dfrdg3  30946