Theorem tfrlem5 5871

Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}

Assertion

Ref	Expression
tfrlem5	⊢ ((g ∈ A ∧ ℎ ∈ A) → ((xgu ∧ xℎv) → u = v))

Distinct variable groups:   f,g,x,y,ℎ,u,v,𝐹   A,g,ℎ

Allowed substitution hints:   A(x,y,v,u,f)

Proof of Theorem tfrlem5

Dummy variables z 𝑎 w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . 3 ⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
2		vex 2554	. . 3 ⊢ g ∈ V
3	1, 2	tfrlem3a 5866	. 2 ⊢ (g ∈ A ↔ ∃z ∈ On (g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))))
4		vex 2554	. . 3 ⊢ ℎ ∈ V
5	1, 4	tfrlem3a 5866	. 2 ⊢ (ℎ ∈ A ↔ ∃w ∈ On (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎))))
6		reeanv 2473	. . 3 ⊢ (∃z ∈ On ∃w ∈ On ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ↔ (∃z ∈ On (g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ ∃w ∈ On (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))))
7		simp2ll 970	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → g Fn z)
8		simp3l 931	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → xgu)
9		fnbr 4944	. . . . . . . . . 10 ⊢ ((g Fn z ∧ xgu) → x ∈ z)
10	7, 8, 9	syl2anc 391	. . . . . . . . 9 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → x ∈ z)
11		simp2rl 972	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → ℎ Fn w)
12		simp3r 932	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → xℎv)
13		fnbr 4944	. . . . . . . . . 10 ⊢ ((ℎ Fn w ∧ xℎv) → x ∈ w)
14	11, 12, 13	syl2anc 391	. . . . . . . . 9 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → x ∈ w)
15		elin 3120	. . . . . . . . 9 ⊢ (x ∈ (z ∩ w) ↔ (x ∈ z ∧ x ∈ w))
16	10, 14, 15	sylanbrc 394	. . . . . . . 8 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → x ∈ (z ∩ w))
17		onin 4089	. . . . . . . . . 10 ⊢ ((z ∈ On ∧ w ∈ On) → (z ∩ w) ∈ On)
18	17	3ad2ant1 924	. . . . . . . . 9 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → (z ∩ w) ∈ On)
19		fnfun 4939	. . . . . . . . . . 11 ⊢ (g Fn z → Fun g)
20	7, 19	syl 14	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → Fun g)
21		inss1 3151	. . . . . . . . . . 11 ⊢ (z ∩ w) ⊆ z
22		fndm 4941	. . . . . . . . . . . 12 ⊢ (g Fn z → dom g = z)
23	7, 22	syl 14	. . . . . . . . . . 11 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → dom g = z)
24	21, 23	syl5sseqr 2988	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → (z ∩ w) ⊆ dom g)
25	20, 24	jca 290	. . . . . . . . 9 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → (Fun g ∧ (z ∩ w) ⊆ dom g))
26		fnfun 4939	. . . . . . . . . . 11 ⊢ (ℎ Fn w → Fun ℎ)
27	11, 26	syl 14	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → Fun ℎ)
28		inss2 3152	. . . . . . . . . . 11 ⊢ (z ∩ w) ⊆ w
29		fndm 4941	. . . . . . . . . . . 12 ⊢ (ℎ Fn w → dom ℎ = w)
30	11, 29	syl 14	. . . . . . . . . . 11 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → dom ℎ = w)
31	28, 30	syl5sseqr 2988	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → (z ∩ w) ⊆ dom ℎ)
32	27, 31	jca 290	. . . . . . . . 9 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → (Fun ℎ ∧ (z ∩ w) ⊆ dom ℎ))
33		simp2lr 971	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎)))
34		ssralv 2998	. . . . . . . . . 10 ⊢ ((z ∩ w) ⊆ z → (∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎)) → ∀𝑎 ∈ (z ∩ w)(g‘𝑎) = (𝐹‘(g ↾ 𝑎))))
35	21, 33, 34	mpsyl 59	. . . . . . . . 9 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → ∀𝑎 ∈ (z ∩ w)(g‘𝑎) = (𝐹‘(g ↾ 𝑎)))
36		simp2rr 973	. . . . . . . . . 10 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))
37		ssralv 2998	. . . . . . . . . 10 ⊢ ((z ∩ w) ⊆ w → (∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)) → ∀𝑎 ∈ (z ∩ w)(ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎))))
38	28, 36, 37	mpsyl 59	. . . . . . . . 9 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → ∀𝑎 ∈ (z ∩ w)(ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))
39	18, 25, 32, 35, 38	tfrlem1 5864	. . . . . . . 8 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → ∀𝑎 ∈ (z ∩ w)(g‘𝑎) = (ℎ‘𝑎))
40		fveq2 5121	. . . . . . . . . 10 ⊢ (𝑎 = x → (g‘𝑎) = (g‘x))
41		fveq2 5121	. . . . . . . . . 10 ⊢ (𝑎 = x → (ℎ‘𝑎) = (ℎ‘x))
42	40, 41	eqeq12d 2051	. . . . . . . . 9 ⊢ (𝑎 = x → ((g‘𝑎) = (ℎ‘𝑎) ↔ (g‘x) = (ℎ‘x)))
43	42	rspcv 2646	. . . . . . . 8 ⊢ (x ∈ (z ∩ w) → (∀𝑎 ∈ (z ∩ w)(g‘𝑎) = (ℎ‘𝑎) → (g‘x) = (ℎ‘x)))
44	16, 39, 43	sylc 56	. . . . . . 7 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → (g‘x) = (ℎ‘x))
45		funbrfv 5155	. . . . . . . 8 ⊢ (Fun g → (xgu → (g‘x) = u))
46	20, 8, 45	sylc 56	. . . . . . 7 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → (g‘x) = u)
47		funbrfv 5155	. . . . . . . 8 ⊢ (Fun ℎ → (xℎv → (ℎ‘x) = v))
48	27, 12, 47	sylc 56	. . . . . . 7 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → (ℎ‘x) = v)
49	44, 46, 48	3eqtr3d 2077	. . . . . 6 ⊢ (((z ∈ On ∧ w ∈ On) ∧ ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) ∧ (xgu ∧ xℎv)) → u = v)
50	49	3exp 1102	. . . . 5 ⊢ ((z ∈ On ∧ w ∈ On) → (((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) → ((xgu ∧ xℎv) → u = v)))
51	50	rexlimdva 2427	. . . 4 ⊢ (z ∈ On → (∃w ∈ On ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) → ((xgu ∧ xℎv) → u = v)))
52	51	rexlimiv 2421	. . 3 ⊢ (∃z ∈ On ∃w ∈ On ((g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) → ((xgu ∧ xℎv) → u = v))
53	6, 52	sylbir 125	. 2 ⊢ ((∃z ∈ On (g Fn z ∧ ∀𝑎 ∈ z (g‘𝑎) = (𝐹‘(g ↾ 𝑎))) ∧ ∃w ∈ On (ℎ Fn w ∧ ∀𝑎 ∈ w (ℎ‘𝑎) = (𝐹‘(ℎ ↾ 𝑎)))) → ((xgu ∧ xℎv) → u = v))
54	3, 5, 53	syl2anb 275	1 ⊢ ((g ∈ A ∧ ℎ ∈ A) → ((xgu ∧ xℎv) → u = v))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301   ∩ cin 2910   ⊆ wss 2911   class class class wbr 3755  Oncon0 4066  dom cdm 4288   ↾ cres 4290  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  tfrlem7  5874  tfrexlem  5889