Theorem tfri3 6382

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 6380). 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 1538	. . . 4 ⊢ Ⅎ𝑥 𝐵 Fn On
2		nfra1 2518	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))
3	1, 2	nfan 1575	. . 3 ⊢ Ⅎ𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)))
4		nfv 1538	. . . . . 6 ⊢ Ⅎ𝑥(𝐵‘𝑦) = (𝐹‘𝑦)
5	3, 4	nfim 1582	. . . . 5 ⊢ Ⅎ𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑦) = (𝐹‘𝑦))
6		fveq2 5527	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵‘𝑥) = (𝐵‘𝑦))
7		fveq2 5527	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
8	6, 7	eqeq12d 2202	. . . . . 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 2564	. . . . . 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 2534	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) → (𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))))
12		onss 4504	. . . . . . . . . . . . . . . . . . 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 6380	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐹 Fn On
16		fvreseq 5632	. . . . . . . . . . . . . . . . . . . . 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 5527	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ↾ 𝑥) = (𝐹 ↾ 𝑥) → (𝐺‘(𝐵 ↾ 𝑥)) = (𝐺‘(𝐹 ↾ 𝑥)))
19	17, 18	syl6bir 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 6381	. . . . . . . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . . . . . . 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 2200	. . . . . . . . . . . . . . . . . 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 4595	. . . 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 2558	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥))
45		eqfnfv 5626	. . . 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 1363   ∈ wcel 2158  ∀wral 2465  Vcvv 2749   ⊆ wss 3141  Oncon0 4375   ↾ cres 4640  Fun wfun 5222   Fn wfn 5223  ‘cfv 5228  recscrecs 6319

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-recs 6320

This theorem is referenced by: (None)