Theorem tfr3 7778

Description: Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. Finally, we show that 𝐹 is unique. We do this by showing that any class 𝐵 with the same properties of 𝐹 that we showed in parts 1 and 2 is identical to 𝐹. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypothesis

Ref	Expression
tfr.1	⊢ 𝐹 = recs(𝐺)

Assertion

Ref	Expression
tfr3	⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺

Proof of Theorem tfr3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1957	. . . 4 ⊢ Ⅎ𝑥 𝐵 Fn On
2		nfra1 3123	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))
3	1, 2	nfan 1946	. . 3 ⊢ Ⅎ𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)))
4		nfv 1957	. . . . . 6 ⊢ Ⅎ𝑥(𝐵‘𝑦) = (𝐹‘𝑦)
5	3, 4	nfim 1943	. . . . 5 ⊢ Ⅎ𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑦) = (𝐹‘𝑦))
6		fveq2 6446	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵‘𝑥) = (𝐵‘𝑦))
7		fveq2 6446	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
8	6, 7	eqeq12d 2793	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐵‘𝑦) = (𝐹‘𝑦)))
9	8	imbi2d 332	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑥) = (𝐹‘𝑥)) ↔ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑦) = (𝐹‘𝑦))))
10		r19.21v 3142	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑦) = (𝐹‘𝑦)) ↔ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)))
11		rsp 3111	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) → (𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))))
12		onss 7268	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
13		tfr.1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐹 = recs(𝐺)
14	13	tfr1 7776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐹 Fn On
15		fvreseq 6582	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 Fn On ∧ 𝐹 Fn On) ∧ 𝑥 ⊆ On) → ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)))
16	14, 15	mpanl2 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 Fn On ∧ 𝑥 ⊆ On) → ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)))
17		fveq2 6446	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))
18	16, 17	syl6bir 246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 Fn On ∧ 𝑥 ⊆ On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
19	12, 18	sylan2 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 Fn On ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
20	19	ancoms 452	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
21	20	imp 397	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))
22	21	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)) ∧ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ 𝑥 ∈ On)) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))
23	13	tfr2 7777	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))
24	23	jctr 520	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))))
25		jcab 513	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On → ((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) ∧ (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))) ↔ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))))
26	24, 25	sylibr 226	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → ((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) ∧ (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))))
27		eqeq12 2791	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) ∧ (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥))) → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
28	26, 27	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))))
29	28	imp 397	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ 𝑥 ∈ On) → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
30	29	adantl 475	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)) ∧ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ 𝑥 ∈ On)) → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
31	22, 30	mpbird 249	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)) ∧ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ 𝑥 ∈ On)) → (𝐵‘𝑥) = (𝐹‘𝑥))
32	31	exp43 429	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → (𝐵‘𝑥) = (𝐹‘𝑥)))))
33	32	com4t 93	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥)))))
34	33	exp4a 424	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥))))))
35	34	pm2.43d 53	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥)))))
36	11, 35	syl 17	. . . . . . . . 9 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥)))))
37	36	com3l 89	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝐵 Fn On → (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥)))))
38	37	impd 400	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥))))
39	38	a2d 29	. . . . . 6 ⊢ (𝑥 ∈ On → (((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)) → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑥) = (𝐹‘𝑥))))
40	10, 39	syl5bi 234	. . . . 5 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑦) = (𝐹‘𝑦)) → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑥) = (𝐹‘𝑥))))
41	5, 9, 40	tfis2f 7333	. . . 4 ⊢ (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑥) = (𝐹‘𝑥)))
42	41	com12 32	. . 3 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → (𝐵‘𝑥) = (𝐹‘𝑥)))
43	3, 42	ralrimi 3139	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥))
44		eqfnfv 6574	. . . 4 ⊢ ((𝐵 Fn On ∧ 𝐹 Fn On) → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥)))
45	14, 44	mpan2 681	. . 3 ⊢ (𝐵 Fn On → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥)))
46	45	biimpar 471	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥)) → 𝐵 = 𝐹)
47	43, 46	syldan 585	1 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090   ⊆ wss 3792   ↾ cres 5357  Oncon0 5976   Fn wfn 6130  ‘cfv 6135  recscrecs 7750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-wrecs 7689  df-recs 7751

This theorem is referenced by: (None)