Theorem tfr3 8438

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 1912	. . . 4 ⊢ Ⅎ𝑥 𝐵 Fn On
2		nfra1 3282	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))
3	1, 2	nfan 1897	. . 3 ⊢ Ⅎ𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)))
4		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑥(𝐵‘𝑦) = (𝐹‘𝑦)
5	3, 4	nfim 1894	. . . . 5 ⊢ Ⅎ𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑦) = (𝐹‘𝑦))
6		fveq2 6907	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵‘𝑥) = (𝐵‘𝑦))
7		fveq2 6907	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
8	6, 7	eqeq12d 2751	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐵‘𝑦) = (𝐹‘𝑦)))
9	8	imbi2d 340	. . . . 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 3178	. . . . . 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 3245	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) → (𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))))
12		onss 7804	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
13		tfr.1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐹 = recs(𝐺)
14	13	tfr1 8436	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐹 Fn On
15		fvreseq 7060	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 Fn On ∧ 𝐹 Fn On) ∧ 𝑥 ⊆ On) → ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)))
16	14, 15	mpanl2 701	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 Fn On ∧ 𝑥 ⊆ On) → ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)))
17		fveq2 6907	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))
18	16, 17	biimtrrdi 254	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 Fn On ∧ 𝑥 ⊆ On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
19	12, 18	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 Fn On ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
20	19	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
21	20	imp 406	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))
22	21	adantr 480	. . . . . . . . . . . . . . 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 8437	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))
24	23	jctr 524	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))))
25		jcab 517	. . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → ((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) ∧ (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))))
27		eqeq12 2752	. . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ 𝑥 ∈ On) → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
30	29	adantl 481	. . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . 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 436	. . . . . . . . . . . . 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 431	. . . . . . . . . . 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 410	. . . . . . 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	biimtrid 242	. . . . 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 7877	. . . 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 3255	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥))
44		eqfnfv 7051	. . . 4 ⊢ ((𝐵 Fn On ∧ 𝐹 Fn On) → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥)))
45	14, 44	mpan2 691	. . 3 ⊢ (𝐵 Fn On → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥)))
46	45	biimpar 477	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥)) → 𝐵 = 𝐹)
47	43, 46	syldan 591	1 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ⊆ wss 3963   ↾ cres 5691  Oncon0 6386   Fn wfn 6558  ‘cfv 6563  recscrecs 8409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410

This theorem is referenced by: (None)