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Theorem tfrlemi14 5869
Description: The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 4-May-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemi14.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
tfrlemi14.2 (Fun 𝐹 (𝐹x) V)
Assertion
Ref Expression
tfrlemi14 dom recs(𝐹) = On
Distinct variable groups:   x,f,y,A   f,𝐹,x,y

Proof of Theorem tfrlemi14
Dummy variables g u w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlemi14.1 . . . 4 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
21tfrlem8 5856 . . 3 Ord dom recs(𝐹)
3 ordsson 4168 . . 3 (Ord dom recs(𝐹) → dom recs(𝐹) ⊆ On)
42, 3ax-mp 7 . 2 dom recs(𝐹) ⊆ On
5 fneq2 4914 . . . . . . . . 9 (w = z → (g Fn wg Fn z))
6 raleq 2483 . . . . . . . . 9 (w = z → (u w (gu) = (𝐹‘(gu)) ↔ u z (gu) = (𝐹‘(gu))))
75, 6anbi12d 445 . . . . . . . 8 (w = z → ((g Fn w u w (gu) = (𝐹‘(gu))) ↔ (g Fn z u z (gu) = (𝐹‘(gu)))))
87exbidv 1688 . . . . . . 7 (w = z → (g(g Fn w u w (gu) = (𝐹‘(gu))) ↔ g(g Fn z u z (gu) = (𝐹‘(gu)))))
9 tru 1232 . . . . . . . 8
10 tfrlemi14.2 . . . . . . . . . . 11 (Fun 𝐹 (𝐹x) V)
1110ax-gen 1318 . . . . . . . . . 10 x(Fun 𝐹 (𝐹x) V)
1211a1i 9 . . . . . . . . 9 ( ⊤ → x(Fun 𝐹 (𝐹x) V))
131, 12tfrlemi1 5867 . . . . . . . 8 (( ⊤ w On) → g(g Fn w u w (gu) = (𝐹‘(gu))))
149, 13mpan 402 . . . . . . 7 (w On → g(g Fn w u w (gu) = (𝐹‘(gu))))
158, 14vtoclga 2596 . . . . . 6 (z On → g(g Fn z u z (gu) = (𝐹‘(gu))))
1611a1i 9 . . . . . . . 8 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → x(Fun 𝐹 (𝐹x) V))
17 simpl 102 . . . . . . . 8 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → z On)
18 simprl 471 . . . . . . . 8 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → g Fn z)
197rspcev 2633 . . . . . . . . 9 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → w On (g Fn w u w (gu) = (𝐹‘(gu))))
20 vex 2538 . . . . . . . . . 10 g V
211, 20tfrlem3a 5847 . . . . . . . . 9 (g Aw On (g Fn w u w (gu) = (𝐹‘(gu))))
2219, 21sylibr 137 . . . . . . . 8 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → g A)
231, 16, 17, 18, 22tfrlemisucaccv 5860 . . . . . . 7 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → (g ∪ {⟨z, (𝐹g)⟩}) A)
24 vex 2538 . . . . . . . . . . 11 z V
2510tfrlem3-2 5849 . . . . . . . . . . . 12 (Fun 𝐹 (𝐹g) V)
2625simpri 106 . . . . . . . . . . 11 (𝐹g) V
2724, 26opex 3940 . . . . . . . . . 10 z, (𝐹g)⟩ V
2827snid 3377 . . . . . . . . 9 z, (𝐹g)⟩ {⟨z, (𝐹g)⟩}
29 elun2 3088 . . . . . . . . 9 (⟨z, (𝐹g)⟩ {⟨z, (𝐹g)⟩} → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
3028, 29mp1i 10 . . . . . . . 8 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
3124, 26opeldm 4465 . . . . . . . 8 (⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}) → z dom (g ∪ {⟨z, (𝐹g)⟩}))
3230, 31syl 14 . . . . . . 7 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → z dom (g ∪ {⟨z, (𝐹g)⟩}))
33 dmeq 4462 . . . . . . . . 9 ( = (g ∪ {⟨z, (𝐹g)⟩}) → dom = dom (g ∪ {⟨z, (𝐹g)⟩}))
3433eleq2d 2089 . . . . . . . 8 ( = (g ∪ {⟨z, (𝐹g)⟩}) → (z dom z dom (g ∪ {⟨z, (𝐹g)⟩})))
3534rspcev 2633 . . . . . . 7 (((g ∪ {⟨z, (𝐹g)⟩}) A z dom (g ∪ {⟨z, (𝐹g)⟩})) → A z dom )
3623, 32, 35syl2anc 393 . . . . . 6 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → A z dom )
3715, 36exlimddv 1760 . . . . 5 (z On → A z dom )
38 eliun 3635 . . . . 5 (z A dom A z dom )
3937, 38sylibr 137 . . . 4 (z On → z A dom )
4039ssriv 2926 . . 3 On ⊆ A dom
411recsfval 5853 . . . . 5 recs(𝐹) = A
4241dmeqi 4463 . . . 4 dom recs(𝐹) = dom A
43 dmuni 4472 . . . 4 dom A = A dom
4442, 43eqtri 2042 . . 3 dom recs(𝐹) = A dom
4540, 44sseqtr4i 2955 . 2 On ⊆ dom recs(𝐹)
464, 45eqssi 2938 1 dom recs(𝐹) = On
Colors of variables: wff set class
Syntax hints:   wa 97  wal 1226   = wceq 1228  wtru 1229  wex 1362   wcel 1374  {cab 2008  wral 2284  wrex 2285  Vcvv 2535  cun 2892  wss 2894  {csn 3350  cop 3353   cuni 3554   ciun 3631  Ord word 4048  Oncon0 4049  dom cdm 4272  cres 4274  Fun wfun 4823   Fn wfn 4824  cfv 4829  recscrecs 5841
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-recs 5842
This theorem is referenced by:  tfri1  5873
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