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Theorem tfrlemi14d 5868
Description: The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)
Hypotheses
Ref Expression
tfrlemi14d.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
tfrlemi14d.2 (φx(Fun 𝐹 (𝐹x) V))
Assertion
Ref Expression
tfrlemi14d (φ → dom recs(𝐹) = On)
Distinct variable groups:   x,f,y,A   f,𝐹,x,y   φ,f,y
Allowed substitution hint:   φ(x)

Proof of Theorem tfrlemi14d
Dummy variables g u w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlemi14d.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, 3mp1i 10 . 2 (φ → dom recs(𝐹) ⊆ On)
5 tfrlemi14d.2 . . . . . . . 8 (φx(Fun 𝐹 (𝐹x) V))
61, 5tfrlemi1 5867 . . . . . . 7 ((φ z On) → g(g Fn z u z (gu) = (𝐹‘(gu))))
75ad2antrr 460 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → x(Fun 𝐹 (𝐹x) V))
8 simplr 470 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → z On)
9 simprl 471 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → g Fn z)
10 fneq2 4914 . . . . . . . . . . . . 13 (w = z → (g Fn wg Fn z))
11 raleq 2483 . . . . . . . . . . . . 13 (w = z → (u w (gu) = (𝐹‘(gu)) ↔ u z (gu) = (𝐹‘(gu))))
1210, 11anbi12d 445 . . . . . . . . . . . 12 (w = z → ((g Fn w u w (gu) = (𝐹‘(gu))) ↔ (g Fn z u z (gu) = (𝐹‘(gu)))))
1312rspcev 2633 . . . . . . . . . . 11 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → w On (g Fn w u w (gu) = (𝐹‘(gu))))
1413adantll 448 . . . . . . . . . 10 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → w On (g Fn w u w (gu) = (𝐹‘(gu))))
15 vex 2538 . . . . . . . . . . 11 g V
161, 15tfrlem3a 5847 . . . . . . . . . 10 (g Aw On (g Fn w u w (gu) = (𝐹‘(gu))))
1714, 16sylibr 137 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → g A)
181, 7, 8, 9, 17tfrlemisucaccv 5860 . . . . . . . 8 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → (g ∪ {⟨z, (𝐹g)⟩}) A)
19 vex 2538 . . . . . . . . . . . 12 z V
205tfrlem3-2d 5850 . . . . . . . . . . . . 13 (φ → (Fun 𝐹 (𝐹g) V))
2120simprd 107 . . . . . . . . . . . 12 (φ → (𝐹g) V)
22 opexg 3938 . . . . . . . . . . . 12 ((z V (𝐹g) V) → ⟨z, (𝐹g)⟩ V)
2319, 21, 22sylancr 395 . . . . . . . . . . 11 (φ → ⟨z, (𝐹g)⟩ V)
24 snidg 3375 . . . . . . . . . . 11 (⟨z, (𝐹g)⟩ V → ⟨z, (𝐹g)⟩ {⟨z, (𝐹g)⟩})
25 elun2 3088 . . . . . . . . . . 11 (⟨z, (𝐹g)⟩ {⟨z, (𝐹g)⟩} → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
2623, 24, 253syl 17 . . . . . . . . . 10 (φ → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
2726ad2antrr 460 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
28 opeldmg 4467 . . . . . . . . . . 11 ((z V (𝐹g) V) → (⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}) → z dom (g ∪ {⟨z, (𝐹g)⟩})))
2919, 21, 28sylancr 395 . . . . . . . . . 10 (φ → (⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}) → z dom (g ∪ {⟨z, (𝐹g)⟩})))
3029ad2antrr 460 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → (⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}) → z dom (g ∪ {⟨z, (𝐹g)⟩})))
3127, 30mpd 13 . . . . . . . 8 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → z dom (g ∪ {⟨z, (𝐹g)⟩}))
32 dmeq 4462 . . . . . . . . . 10 ( = (g ∪ {⟨z, (𝐹g)⟩}) → dom = dom (g ∪ {⟨z, (𝐹g)⟩}))
3332eleq2d 2089 . . . . . . . . 9 ( = (g ∪ {⟨z, (𝐹g)⟩}) → (z dom z dom (g ∪ {⟨z, (𝐹g)⟩})))
3433rspcev 2633 . . . . . . . 8 (((g ∪ {⟨z, (𝐹g)⟩}) A z dom (g ∪ {⟨z, (𝐹g)⟩})) → A z dom )
3518, 31, 34syl2anc 393 . . . . . . 7 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → A z dom )
366, 35exlimddv 1760 . . . . . 6 ((φ z On) → A z dom )
37 eliun 3635 . . . . . 6 (z A dom A z dom )
3836, 37sylibr 137 . . . . 5 ((φ z On) → z A dom )
3938ex 108 . . . 4 (φ → (z On → z A dom ))
4039ssrdv 2928 . . 3 (φ → 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, 44syl6sseqr 2969 . 2 (φ → On ⊆ dom recs(𝐹))
464, 45eqssd 2939 1 (φ → dom recs(𝐹) = On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226   = wceq 1228   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:  tfri1d  5871
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