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Theorem tfr3 7648
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 1995 . . . 4 𝑥 𝐵 Fn On
2 nfra1 3090 . . . 4 𝑥𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))
31, 2nfan 1980 . . 3 𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)))
4 nfv 1995 . . . . . 6 𝑥(𝐵𝑦) = (𝐹𝑦)
53, 4nfim 1977 . . . . 5 𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑦) = (𝐹𝑦))
6 fveq2 6332 . . . . . . 7 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
7 fveq2 6332 . . . . . . 7 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
86, 7eqeq12d 2786 . . . . . 6 (𝑥 = 𝑦 → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐵𝑦) = (𝐹𝑦)))
98imbi2d 329 . . . . 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 3109 . . . . . 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 3078 . . . . . . . . . 10 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))))
12 onss 7137 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → 𝑥 ⊆ On)
13 tfr.1 . . . . . . . . . . . . . . . . . . . . . 22 𝐹 = recs(𝐺)
1413tfr1 7646 . . . . . . . . . . . . . . . . . . . . 21 𝐹 Fn On
15 fvreseq 6462 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 Fn On ∧ 𝐹 Fn On) ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
1614, 15mpanl2 681 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 Fn On ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
17 fveq2 6332 . . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . 18 ((𝐵 Fn On ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2019ancoms 455 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2120imp 393 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2221adantr 466 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2313tfr2 7647 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))
2423jctr 514 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
25 jcab 507 . . . . . . . . . . . . . . . . . . 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 2784 . . . . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
3029adantl 467 . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . 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 418 . . . . . . . . . . 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 396 . . . . . . 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 7202 . . . 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 3106 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥))
44 eqfnfv 6454 . . . 4 ((𝐵 Fn On ∧ 𝐹 Fn On) → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4514, 44mpan2 671 . . 3 (𝐵 Fn On → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4645biimpar 463 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)) → 𝐵 = 𝐹)
4743, 46syldan 579 1 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wss 3723  cres 5251  Oncon0 5866   Fn wfn 6026  cfv 6031  recscrecs 7620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-wrecs 7559  df-recs 7621
This theorem is referenced by: (None)
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