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Theorem tfrlemibacc 5861
Description: Each element of B is an acceptable function. Lemma for tfrlemi1 5867. (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 900 . . . . . . 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 460 . . . . . . . 8 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → x(Fun 𝐹 (𝐹x) V))
6 tfrlemi1.4 . . . . . . . . . 10 (φx On)
76ad2antrr 460 . . . . . . . . 9 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → x On)
8 simplr 470 . . . . . . . . 9 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → z x)
9 onelon 4070 . . . . . . . . 9 ((x On z x) → z On)
107, 8, 9syl2anc 393 . . . . . . . 8 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → z On)
11 simpr1 898 . . . . . . . 8 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → g Fn z)
12 simpr2 899 . . . . . . . 8 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → g A)
133, 5, 10, 11, 12tfrlemisucaccv 5860 . . . . . . 7 (((φ z x) (g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))) → (g ∪ {⟨z, (𝐹g)⟩}) A)
142, 13eqeltrd 2096 . . . . . 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 1682 . . . 4 ((φ z x) → (g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → A))
1716rexlimdva 2411 . . 3 (φ → (z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩})) → A))
1817abssdv 2991 . 2 (φ → {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))} ⊆ A)
191, 18syl5eqss 2966 1 (φBA)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873  wal 1226   = wceq 1228  wex 1362   wcel 1374  {cab 2008  wral 2284  wrex 2285  Vcvv 2535  cun 2892  wss 2894  {csn 3350  cop 3353  Oncon0 4049  cres 4274  Fun wfun 4823   Fn wfn 4824  cfv 4829
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-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-v 2537  df-sbc 2742  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-br 3739  df-opab 3793  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-res 4284  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  tfrlemibfn  5863  tfrlemiubacc  5865
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