Theorem tfri3 6576

Description: Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 6574). 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 Jim Kingdon, 4-May-2019.)

Hypotheses

Ref	Expression
tfri3.1	⊢ 𝐹 = recs(𝐺)
tfri3.2	⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)

Assertion

Ref	Expression
tfri3	⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)

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

Proof of Theorem tfri3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1577	. . . 4 ⊢ Ⅎ𝑥 𝐵 Fn On
2		nfra1 2564	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))
3	1, 2	nfan 1614	. . 3 ⊢ Ⅎ𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)))
4		nfv 1577	. . . . . 6 ⊢ Ⅎ𝑥(𝐵‘𝑦) = (𝐹‘𝑦)
5	3, 4	nfim 1621	. . . . 5 ⊢ Ⅎ𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑦) = (𝐹‘𝑦))
6		fveq2 5648	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵‘𝑥) = (𝐵‘𝑦))
7		fveq2 5648	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
8	6, 7	eqeq12d 2246	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐵‘𝑦) = (𝐹‘𝑦)))
9	8	imbi2d 230	. . . . 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 2610	. . . . . 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 2580	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) → (𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))))
12		onss 4597	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
13		tfri3.1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐹 = recs(𝐺)
14		tfri3.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)
15	13, 14	tfri1 6574	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐹 Fn On
16		fvreseq 5759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 Fn On ∧ 𝐹 Fn On) ∧ 𝑥 ⊆ On) → ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)))
17	15, 16	mpanl2 435	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 Fn On ∧ 𝑥 ⊆ On) → ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) ↔ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)))
18		fveq2 5648	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))
19	17, 18	biimtrrdi 164	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 Fn On ∧ 𝑥 ⊆ On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
20	12, 19	sylan2 286	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 Fn On ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
21	20	ancoms 268	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
22	21	imp 124	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦)) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))
23	22	adantr 276	. . . . . . . . . . . . . . 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)))
24	13, 14	tfri2 6575	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))
25	24	jctr 315	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))))
26		jcab 607	. . . . . . . . . . . . . . . . . . 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)))))
27	25, 26	sylibr 134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → ((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) ∧ (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))))
28		eqeq12 2244	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) ∧ (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥))) → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
29	27, 28	syl6 33	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))))
30	29	imp 124	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) ∧ 𝑥 ∈ On) → ((𝐵‘𝑥) = (𝐹‘𝑥) ↔ (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥))))
31	30	adantl 277	. . . . . . . . . . . . . . 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))))
32	23, 31	mpbird 167	. . . . . . . . . . . . . 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))
33	32	exp43 372	. . . . . . . . . . . . 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)))))
34	33	com4t 85	. . . . . . . . . . . 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)))))
35	34	exp4a 366	. . . . . . . . . . 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))))))
36	35	pm2.43d 50	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥)))))
37	11, 36	syl 14	. . . . . . . . 9 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥)))))
38	37	com3l 81	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝐵 Fn On → (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥)))))
39	38	impd 254	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐹‘𝑦) → (𝐵‘𝑥) = (𝐹‘𝑥))))
40	39	a2d 26	. . . . . 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))))
41	10, 40	biimtrid 152	. . . . 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))))
42	5, 9, 41	tfis2f 4688	. . . 4 ⊢ (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑥) = (𝐹‘𝑥)))
43	42	com12 30	. . 3 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝑥 ∈ On → (𝐵‘𝑥) = (𝐹‘𝑥)))
44	3, 43	ralrimi 2604	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥))
45		eqfnfv 5753	. . . 4 ⊢ ((𝐵 Fn On ∧ 𝐹 Fn On) → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥)))
46	15, 45	mpan2 425	. . 3 ⊢ (𝐵 Fn On → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥)))
47	46	biimpar 297	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥)) → 𝐵 = 𝐹)
48	44, 47	syldan 282	1 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803   ⊆ wss 3201  Oncon0 4466   ↾ cres 4733  Fun wfun 5327   Fn wfn 5328  ‘cfv 5333  recscrecs 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-recs 6514

This theorem is referenced by: (None)