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Theorem tfri3 5953
 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 5951). 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 1421 . . . 4 𝑥 𝐵 Fn On
2 nfra1 2355 . . . 4 𝑥𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))
31, 2nfan 1457 . . 3 𝑥(𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)))
4 nfv 1421 . . . . . 6 𝑥(𝐵𝑦) = (𝐹𝑦)
53, 4nfim 1464 . . . . 5 𝑥((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑦) = (𝐹𝑦))
6 fveq2 5178 . . . . . . 7 (𝑥 = 𝑦 → (𝐵𝑥) = (𝐵𝑦))
7 fveq2 5178 . . . . . . 7 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
86, 7eqeq12d 2054 . . . . . 6 (𝑥 = 𝑦 → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐵𝑦) = (𝐹𝑦)))
98imbi2d 219 . . . . 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 2396 . . . . . 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 2369 . . . . . . . . . 10 (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))))
12 onss 4219 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → 𝑥 ⊆ On)
13 tfri3.1 . . . . . . . . . . . . . . . . . . . . . 22 𝐹 = recs(𝐺)
14 tfri3.2 . . . . . . . . . . . . . . . . . . . . . 22 (Fun 𝐺 ∧ (𝐺𝑥) ∈ V)
1513, 14tfri1 5951 . . . . . . . . . . . . . . . . . . . . 21 𝐹 Fn On
16 fvreseq 5271 . . . . . . . . . . . . . . . . . . . . 21 (((𝐵 Fn On ∧ 𝐹 Fn On) ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
1715, 16mpanl2 411 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 Fn On ∧ 𝑥 ⊆ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)))
18 fveq2 5178 . . . . . . . . . . . . . . . . . . . 20 ((𝐵𝑥) = (𝐹𝑥) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
1917, 18syl6bir 153 . . . . . . . . . . . . . . . . . . 19 ((𝐵 Fn On ∧ 𝑥 ⊆ On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2012, 19sylan2 270 . . . . . . . . . . . . . . . . . 18 ((𝐵 Fn On ∧ 𝑥 ∈ On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2120ancoms 255 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2221imp 115 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2322adantr 261 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))
2413, 14tfri2 5952 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))
2524jctr 298 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
26 jcab 535 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥)))) ↔ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
2725, 26sylibr 137 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥)))))
28 eqeq12 2052 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑥) = (𝐺‘(𝐵𝑥)) ∧ (𝐹𝑥) = (𝐺‘(𝐹𝑥))) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
2927, 28syl6 29 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥)))))
3029imp 115 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
3130adantl 262 . . . . . . . . . . . . . . 15 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → ((𝐵𝑥) = (𝐹𝑥) ↔ (𝐺‘(𝐵𝑥)) = (𝐺‘(𝐹𝑥))))
3223, 31mpbird 156 . . . . . . . . . . . . . 14 ((((𝑥 ∈ On ∧ 𝐵 Fn On) ∧ ∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦)) ∧ ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) ∧ 𝑥 ∈ On)) → (𝐵𝑥) = (𝐹𝑥))
3332exp43 354 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝐵𝑥) = (𝐹𝑥)))))
3433com4t 79 . . . . . . . . . . . 12 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → ((𝑥 ∈ On ∧ 𝐵 Fn On) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3534exp4a 348 . . . . . . . . . . 11 ((𝑥 ∈ On → (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝑥 ∈ On → (𝐵 Fn On → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥))))))
3635pm2.43d 44 . . . . . . . . . 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 75 . . . . . . . 8 (𝑥 ∈ On → (𝐵 Fn On → (∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥)) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥)))))
3938impd 242 . . . . . . 7 (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (∀𝑦𝑥 (𝐵𝑦) = (𝐹𝑦) → (𝐵𝑥) = (𝐹𝑥))))
4039a2d 23 . . . . . 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 141 . . . . 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 4307 . . . 4 (𝑥 ∈ On → ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝐵𝑥) = (𝐹𝑥)))
4342com12 27 . . 3 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → (𝑥 ∈ On → (𝐵𝑥) = (𝐹𝑥)))
443, 43ralrimi 2390 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥))
45 eqfnfv 5265 . . . 4 ((𝐵 Fn On ∧ 𝐹 Fn On) → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4615, 45mpan2 401 . . 3 (𝐵 Fn On → (𝐵 = 𝐹 ↔ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)))
4746biimpar 281 . 2 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐹𝑥)) → 𝐵 = 𝐹)
4844, 47syldan 266 1 ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵𝑥) = (𝐺‘(𝐵𝑥))) → 𝐵 = 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306  Vcvv 2557   ⊆ wss 2917  Oncon0 4100   ↾ cres 4347  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902  recscrecs 5919 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920 This theorem is referenced by: (None)
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