Theorem tfri3 6371

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 6369). 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 1528	. . . 4 ⊢ Ⅎ𝑥 𝐵 Fn On
2		nfra1 2508	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))
3	1, 2	nfan 1565	. . 3 ⊢ Ⅎ𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)))
4		nfv 1528	. . . . . 6 ⊢ Ⅎ𝑥(𝐵‘𝑦) = (𝐹‘𝑦)
5	3, 4	nfim 1572	. . . . 5 ⊢ Ⅎ𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → (𝐵‘𝑦) = (𝐹‘𝑦))
6		fveq2 5517	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵‘𝑥) = (𝐵‘𝑦))
7		fveq2 5517	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
8	6, 7	eqeq12d 2192	. . . . . 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 2554	. . . . . 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 2524	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)) → (𝑥 ∈ On → (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))))
12		onss 4494	. . . . . . . . . . . . . . . . . . 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 6369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐹 Fn On
16		fvreseq 5622	. . . . . . . . . . . . . . . . . . . . 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 5517	. . . . . . . . . . . . . . . . . . . 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 6370	. . . . . . . . . . . . . . . . . . . 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 2190	. . . . . . . . . . . . . . . . . 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 4585	. . . 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 2548	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐹‘𝑥))
45		eqfnfv 5616	. . . 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 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2739   ⊆ wss 3131  Oncon0 4365   ↾ cres 4630  Fun wfun 5212   Fn wfn 5213  ‘cfv 5218  recscrecs 6308

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6309

This theorem is referenced by: (None)