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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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