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Theorem tfrlemiex 5886
 Description: Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 18-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
tfrlemiex (φf(f Fn x u x (fu) = (𝐹‘(fu))))
Distinct variable groups:   f,g,,u,w,x,y,z,A   f,𝐹,g,,u,w,x,y,z   φ,w,y   u,B,w,f,g,,z   φ,g,,z
Allowed substitution hints:   φ(x,u,f)   B(x,y)

Proof of Theorem tfrlemiex
StepHypRef Expression
1 tfrlemisucfn.1 . . . 4 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
2 tfrlemisucfn.2 . . . 4 (φx(Fun 𝐹 (𝐹x) V))
3 tfrlemi1.3 . . . 4 B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}
4 tfrlemi1.4 . . . 4 (φx On)
5 tfrlemi1.5 . . . 4 (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))
61, 2, 3, 4, 5tfrlemibex 5884 . . 3 (φB V)
7 uniexg 4141 . . 3 (B V → B V)
86, 7syl 14 . 2 (φ B V)
91, 2, 3, 4, 5tfrlemibfn 5883 . . 3 (φ B Fn x)
101, 2, 3, 4, 5tfrlemiubacc 5885 . . 3 (φu x ( Bu) = (𝐹‘( Bu)))
119, 10jca 290 . 2 (φ → ( B Fn x u x ( Bu) = (𝐹‘( Bu))))
12 fneq1 4930 . . . 4 (f = B → (f Fn x B Fn x))
13 fveq1 5120 . . . . . 6 (f = B → (fu) = ( Bu))
14 reseq1 4549 . . . . . . 7 (f = B → (fu) = ( Bu))
1514fveq2d 5125 . . . . . 6 (f = B → (𝐹‘(fu)) = (𝐹‘( Bu)))
1613, 15eqeq12d 2051 . . . . 5 (f = B → ((fu) = (𝐹‘(fu)) ↔ ( Bu) = (𝐹‘( Bu))))
1716ralbidv 2320 . . . 4 (f = B → (u x (fu) = (𝐹‘(fu)) ↔ u x ( Bu) = (𝐹‘( Bu))))
1812, 17anbi12d 442 . . 3 (f = B → ((f Fn x u x (fu) = (𝐹‘(fu))) ↔ ( B Fn x u x ( Bu) = (𝐹‘( Bu)))))
1918spcegv 2635 . 2 ( B V → (( B Fn x u x ( Bu) = (𝐹‘( Bu))) → f(f Fn x u x (fu) = (𝐹‘(fu)))))
208, 11, 19sylc 56 1 (φf(f Fn x u x (fu) = (𝐹‘(fu))))
 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  {csn 3367  ⟨cop 3370  ∪ cuni 3571  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-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:  tfrlemi1  5887
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