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Theorem tfrlemibacc 5881
Description: Each element of B is an acceptable function. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
tfrlemisucfn.2 (φx(Fun 𝐹 (𝐹x) V))
tfrlemi1.3 B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}
tfrlemi1.4 (φx On)
tfrlemi1.5 (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))
Assertion
Ref Expression
tfrlemibacc (φBA)
Distinct variable groups:   f,g,,w,x,y,z,A   f,𝐹,g,,w,x,y,z   φ,w,y   w,B,f,g,,z   φ,g,,z
Allowed substitution hints:   φ(x,f)   B(x,y)

Proof of Theorem tfrlemibacc
StepHypRef Expression
1 tfrlemi1.3 . 2 B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}
2 simpr3 911 . . . . . . 7 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → = (g ∪ {⟨z, (𝐹g)⟩}))
3 tfrlemisucfn.1 . . . . . . . 8 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
4 tfrlemisucfn.2 . . . . . . . . 9 (φx(Fun 𝐹 (𝐹x) V))
54ad2antrr 457 . . . . . . . 8 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → x(Fun 𝐹 (𝐹x) V))
6 tfrlemi1.4 . . . . . . . . . 10 (φx On)
76ad2antrr 457 . . . . . . . . 9 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → x On)
8 simplr 482 . . . . . . . . 9 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → z x)
9 onelon 4087 . . . . . . . . 9 ((x On z x) → z On)
107, 8, 9syl2anc 391 . . . . . . . 8 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → z On)
11 simpr1 909 . . . . . . . 8 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → g Fn z)
12 simpr2 910 . . . . . . . 8 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → g A)
133, 5, 10, 11, 12tfrlemisucaccv 5880 . . . . . . 7 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) A)
142, 13eqeltrd 2111 . . . . . 6 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → A)
1514ex 108 . . . . 5 ((φ z x) → ((g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → A))
1615exlimdv 1697 . . . 4 ((φ z x) → (g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → A))
1716rexlimdva 2427 . . 3 (φ → (z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → A))
1817abssdv 3008 . 2 (φ → {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))} ⊆ A)
191, 18syl5eqss 2983 1 (φBA)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884  wal 1240   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  cun 2909  wss 2911  {csn 3367  cop 3370  Oncon0 4066  cres 4290  Fun wfun 4839   Fn wfn 4840  cfv 4845
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-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-v 2553  df-sbc 2759  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-br 3756  df-opab 3810  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-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  tfrlemibfn  5883  tfrlemiubacc  5885
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