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Theorem tfr3 7441
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 1845 . . . 4 𝑥 𝐵 Fn On
2 nfra1 2941 . . . 4 𝑥𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))
31, 2nfan 1830 . . 3 𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)))
4 nfv 1845 . . . . . 6 𝑥(𝐵𝑦) = (𝐹𝑦)
53, 4nfim 1827 . . . . 5 𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑦) = (𝐹𝑦))
6 fveq2 6150 . . . . . . 7 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
7 fveq2 6150 . . . . . . 7 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
86, 7eqeq12d 2641 . . . . . 6 (𝑥 = 𝑦 → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐵𝑦) = (𝐹𝑦)))
98imbi2d 330 . . . . 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 2959 . . . . . 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 2929 . . . . . . . . . 10 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))))
12 onss 6938 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → 𝑥 ⊆ On)
13 tfr.1 . . . . . . . . . . . . . . . . . . . . . 22 𝐹 = recs(𝐺)
1413tfr1 7439 . . . . . . . . . . . . . . . . . . . . 21 𝐹 Fn On
15 fvreseq 6276 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 Fn On ∧ 𝐹 Fn On) ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
1614, 15mpanl2 716 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 Fn On ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
17 fveq2 6150 . . . . . . . . . . . . . . . . . . . 20 ((𝐵𝑥) = (𝐹𝑥) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
1816, 17syl6bir 244 . . . . . . . . . . . . . . . . . . 19 ((𝐵 Fn On ∧ 𝑥 ⊆ On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
1912, 18sylan2 491 . . . . . . . . . . . . . . . . . 18 ((𝐵 Fn On ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2019ancoms 469 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2120imp 445 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2221adantr 481 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2313tfr2 7440 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))
2423jctr 564 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
25 jcab 906 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥)))) ↔ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
2624, 25sylibr 224 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
27 eqeq12 2639 . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
3029adantl 482 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
3122, 30mpbird 247 . . . . . . . . . . . . . 14 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → (𝐵𝑥) = (𝐹𝑥))
3231exp43 639 . . . . . . . . . . . . 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 632 . . . . . . . . . . 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 447 . . . . . . 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 232 . . . . 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 7003 . . . 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 2956 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥))
44 eqfnfv 6268 . . . 4 ((𝐵 Fn On ∧ 𝐹 Fn On) → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4514, 44mpan2 706 . . 3 (𝐵 Fn On → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4645biimpar 502 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)) → 𝐵 = 𝐹)
4743, 46syldan 487 1 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wss 3560  cres 5081  Oncon0 5685   Fn wfn 5845  cfv 5850  recscrecs 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-wrecs 7353  df-recs 7414
This theorem is referenced by: (None)
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