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Theorem tfri3 6316
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 6314). 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 ∧ (𝐺𝑥) ∈ V)
Assertion
Ref Expression
tfri3 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺

Proof of Theorem tfri3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 1508 . . . 4 𝑥 𝐵 Fn On
2 nfra1 2488 . . . 4 𝑥𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))
31, 2nfan 1545 . . 3 𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)))
4 nfv 1508 . . . . . 6 𝑥(𝐵𝑦) = (𝐹𝑦)
53, 4nfim 1552 . . . . 5 𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑦) = (𝐹𝑦))
6 fveq2 5470 . . . . . . 7 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
7 fveq2 5470 . . . . . . 7 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
86, 7eqeq12d 2172 . . . . . 6 (𝑥 = 𝑦 → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐵𝑦) = (𝐹𝑦)))
98imbi2d 229 . . . . 5 (𝑥 = 𝑦 → (((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑥) = (𝐹𝑥)) ↔ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑦) = (𝐹𝑦))))
10 r19.21v 2534 . . . . . 6 (∀𝑦𝑥 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑦) = (𝐹𝑦)) ↔ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
11 rsp 2504 . . . . . . . . . 10 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))))
12 onss 4454 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → 𝑥 ⊆ On)
13 tfri3.1 . . . . . . . . . . . . . . . . . . . . . 22 𝐹 = recs(𝐺)
14 tfri3.2 . . . . . . . . . . . . . . . . . . . . . 22 (Fun 𝐺 ∧ (𝐺𝑥) ∈ V)
1513, 14tfri1 6314 . . . . . . . . . . . . . . . . . . . . 21 𝐹 Fn On
16 fvreseq 5573 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 Fn On ∧ 𝐹 Fn On) ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
1715, 16mpanl2 432 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 Fn On ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
18 fveq2 5470 . . . . . . . . . . . . . . . . . . . 20 ((𝐵𝑥) = (𝐹𝑥) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
1917, 18syl6bir 163 . . . . . . . . . . . . . . . . . . 19 ((𝐵 Fn On ∧ 𝑥 ⊆ On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2012, 19sylan2 284 . . . . . . . . . . . . . . . . . 18 ((𝐵 Fn On ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2120ancoms 266 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2221imp 123 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2322adantr 274 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2413, 14tfri2 6315 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))
2524jctr 313 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
26 jcab 593 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥)))) ↔ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
2725, 26sylibr 133 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
28 eqeq12 2170 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥))) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2927, 28syl6 33 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))))
3029imp 123 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
3130adantl 275 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
3223, 31mpbird 166 . . . . . . . . . . . . . 14 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → (𝐵𝑥) = (𝐹𝑥))
3332exp43 370 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝐵𝑥) = (𝐹𝑥)))))
3433com4t 85 . . . . . . . . . . . 12 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3534exp4a 364 . . . . . . . . . . 11 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥))))))
3635pm2.43d 50 . . . . . . . . . 10 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3711, 36syl 14 . . . . . . . . 9 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3837com3l 81 . . . . . . . 8 (𝑥 ∈ On → (𝐵 Fn On → (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3938impd 252 . . . . . . 7 (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥))))
4039a2d 26 . . . . . 6 (𝑥 ∈ On → (((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑥) = (𝐹𝑥))))
4110, 40syl5bi 151 . . . . 5 (𝑥 ∈ On → (∀𝑦𝑥 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑦) = (𝐹𝑦)) → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑥) = (𝐹𝑥))))
425, 9, 41tfis2f 4545 . . . 4 (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑥) = (𝐹𝑥)))
4342com12 30 . . 3 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝐵𝑥) = (𝐹𝑥)))
443, 43ralrimi 2528 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥))
45 eqfnfv 5567 . . . 4 ((𝐵 Fn On ∧ 𝐹 Fn On) → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4615, 45mpan2 422 . . 3 (𝐵 Fn On → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4746biimpar 295 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)) → 𝐵 = 𝐹)
4844, 47syldan 280 1 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wcel 2128  wral 2435  Vcvv 2712  wss 3102  Oncon0 4325  cres 4590  Fun wfun 5166   Fn wfn 5167  cfv 5172  recscrecs 6253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-recs 6254
This theorem is referenced by: (None)
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