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Theorem tfrlemi14d 5888
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 5875 . . 3 Ord dom recs(𝐹)
3 ordsson 4184 . . 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 5887 . . . . . . 7 ((φ z On) → g(g Fn z u z (gu) = (𝐹‘(gu))))
75ad2antrr 457 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → x(Fun 𝐹 (𝐹x) V))
8 simplr 482 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → z On)
9 simprl 483 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → g Fn z)
10 fneq2 4931 . . . . . . . . . . . . 13 (w = z → (g Fn wg Fn z))
11 raleq 2499 . . . . . . . . . . . . 13 (w = z → (u w (gu) = (𝐹‘(gu)) ↔ u z (gu) = (𝐹‘(gu))))
1210, 11anbi12d 442 . . . . . . . . . . . 12 (w = z → ((g Fn w u w (gu) = (𝐹‘(gu))) ↔ (g Fn z u z (gu) = (𝐹‘(gu)))))
1312rspcev 2650 . . . . . . . . . . 11 ((z On (g Fn z u z (gu) = (𝐹‘(gu)))) → w On (g Fn w u w (gu) = (𝐹‘(gu))))
1413adantll 445 . . . . . . . . . 10 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → w On (g Fn w u w (gu) = (𝐹‘(gu))))
15 vex 2554 . . . . . . . . . . 11 g V
161, 15tfrlem3a 5866 . . . . . . . . . 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 5880 . . . . . . . 8 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → (g ∪ {⟨z, (𝐹g)⟩}) A)
19 vex 2554 . . . . . . . . . . . 12 z V
205tfrlem3-2d 5869 . . . . . . . . . . . . 13 (φ → (Fun 𝐹 (𝐹g) V))
2120simprd 107 . . . . . . . . . . . 12 (φ → (𝐹g) V)
22 opexg 3955 . . . . . . . . . . . 12 ((z V (𝐹g) V) → ⟨z, (𝐹g)⟩ V)
2319, 21, 22sylancr 393 . . . . . . . . . . 11 (φ → ⟨z, (𝐹g)⟩ V)
24 snidg 3392 . . . . . . . . . . 11 (⟨z, (𝐹g)⟩ V → ⟨z, (𝐹g)⟩ {⟨z, (𝐹g)⟩})
25 elun2 3105 . . . . . . . . . . 11 (⟨z, (𝐹g)⟩ {⟨z, (𝐹g)⟩} → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
2623, 24, 253syl 17 . . . . . . . . . 10 (φ → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
2726ad2antrr 457 . . . . . . . . 9 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → ⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}))
28 opeldmg 4483 . . . . . . . . . . 11 ((z V (𝐹g) V) → (⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}) → z dom (g ∪ {⟨z, (𝐹g)⟩})))
2919, 21, 28sylancr 393 . . . . . . . . . 10 (φ → (⟨z, (𝐹g)⟩ (g ∪ {⟨z, (𝐹g)⟩}) → z dom (g ∪ {⟨z, (𝐹g)⟩})))
3029ad2antrr 457 . . . . . . . . 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 4478 . . . . . . . . . 10 ( = (g ∪ {⟨z, (𝐹g)⟩}) → dom = dom (g ∪ {⟨z, (𝐹g)⟩}))
3332eleq2d 2104 . . . . . . . . 9 ( = (g ∪ {⟨z, (𝐹g)⟩}) → (z dom z dom (g ∪ {⟨z, (𝐹g)⟩})))
3433rspcev 2650 . . . . . . . 8 (((g ∪ {⟨z, (𝐹g)⟩}) A z dom (g ∪ {⟨z, (𝐹g)⟩})) → A z dom )
3518, 31, 34syl2anc 391 . . . . . . 7 (((φ z On) (g Fn z u z (gu) = (𝐹‘(gu)))) → A z dom )
366, 35exlimddv 1775 . . . . . 6 ((φ z On) → A z dom )
37 eliun 3652 . . . . . 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 2945 . . 3 (φ → On ⊆ A dom )
411recsfval 5872 . . . . 5 recs(𝐹) = A
4241dmeqi 4479 . . . 4 dom recs(𝐹) = dom A
43 dmuni 4488 . . . 4 dom A = A dom
4442, 43eqtri 2057 . . 3 dom recs(𝐹) = A dom
4540, 44syl6sseqr 2986 . 2 (φ → On ⊆ dom recs(𝐹))
464, 45eqssd 2956 1 (φ → dom recs(𝐹) = On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  cun 2909  wss 2911  {csn 3367  cop 3370   cuni 3571   ciun 3648  Ord word 4065  Oncon0 4066  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840  cfv 4845  recscrecs 5860
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861
This theorem is referenced by:  tfri1d  5890
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