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Theorem tfr3 8031
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 (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺

Proof of Theorem tfr3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 1916 . . . 4 𝑥 𝐵 Fn On
2 nfra1 3213 . . . 4 𝑥𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))
31, 2nfan 1901 . . 3 𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)))
4 nfv 1916 . . . . . 6 𝑥(𝐵𝑦) = (𝐹𝑦)
53, 4nfim 1898 . . . . 5 𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑦) = (𝐹𝑦))
6 fveq2 6661 . . . . . . 7 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
7 fveq2 6661 . . . . . . 7 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
86, 7eqeq12d 2840 . . . . . 6 (𝑥 = 𝑦 → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐵𝑦) = (𝐹𝑦)))
98imbi2d 344 . . . . 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 3170 . . . . . 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 3200 . . . . . . . . . 10 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))))
12 onss 7499 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → 𝑥 ⊆ On)
13 tfr.1 . . . . . . . . . . . . . . . . . . . . . 22 𝐹 = recs(𝐺)
1413tfr1 8029 . . . . . . . . . . . . . . . . . . . . 21 𝐹 Fn On
15 fvreseq 6801 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 Fn On ∧ 𝐹 Fn On) ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
1614, 15mpanl2 700 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 Fn On ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
17 fveq2 6661 . . . . . . . . . . . . . . . . . . . 20 ((𝐵𝑥) = (𝐹𝑥) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
1816, 17syl6bir 257 . . . . . . . . . . . . . . . . . . 19 ((𝐵 Fn On ∧ 𝑥 ⊆ On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
1912, 18sylan2 595 . . . . . . . . . . . . . . . . . 18 ((𝐵 Fn On ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2019ancoms 462 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2120imp 410 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2221adantr 484 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2313tfr2 8030 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))
2423jctr 528 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
25 jcab 521 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥)))) ↔ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
2624, 25sylibr 237 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
27 eqeq12 2838 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥))) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2826, 27syl6 35 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))))
2928imp 410 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
3029adantl 485 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
3122, 30mpbird 260 . . . . . . . . . . . . . 14 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → (𝐵𝑥) = (𝐹𝑥))
3231exp43 440 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝐵𝑥) = (𝐹𝑥)))))
3332com4t 93 . . . . . . . . . . . 12 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3433exp4a 435 . . . . . . . . . . 11 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥))))))
3534pm2.43d 53 . . . . . . . . . 10 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3611, 35syl 17 . . . . . . . . 9 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3736com3l 89 . . . . . . . 8 (𝑥 ∈ On → (𝐵 Fn On → (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3837impd 414 . . . . . . 7 (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥))))
3938a2d 29 . . . . . 6 (𝑥 ∈ On → (((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑥) = (𝐹𝑥))))
4010, 39syl5bi 245 . . . . 5 (𝑥 ∈ On → (∀𝑦𝑥 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑦) = (𝐹𝑦)) → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑥) = (𝐹𝑥))))
415, 9, 40tfis2f 7564 . . . 4 (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑥) = (𝐹𝑥)))
4241com12 32 . . 3 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝐵𝑥) = (𝐹𝑥)))
433, 42ralrimi 3210 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥))
44 eqfnfv 6793 . . . 4 ((𝐵 Fn On ∧ 𝐹 Fn On) → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4514, 44mpan2 690 . . 3 (𝐵 Fn On → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4645biimpar 481 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)) → 𝐵 = 𝐹)
4743, 46syldan 594 1 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  wss 3919  cres 5544  Oncon0 6178   Fn wfn 6338  cfv 6343  recscrecs 8003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-wrecs 7943  df-recs 8004
This theorem is referenced by: (None)
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