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Theorem fvmptdv2 5162
Description: Alternate deduction version of fvmpt 5151, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdv2.1 (φA 𝐷)
fvmptdv2.2 ((φ x = A) → B 𝑉)
fvmptdv2.3 ((φ x = A) → B = 𝐶)
Assertion
Ref Expression
fvmptdv2 (φ → (𝐹 = (x 𝐷B) → (𝐹A) = 𝐶))
Distinct variable groups:   x,A   x,𝐶   x,𝐷   φ,x
Allowed substitution hints:   B(x)   𝐹(x)   𝑉(x)

Proof of Theorem fvmptdv2
StepHypRef Expression
1 eqidd 2024 . . 3 (φ → (x 𝐷B) = (x 𝐷B))
2 fvmptdv2.3 . . 3 ((φ x = A) → B = 𝐶)
3 fvmptdv2.1 . . 3 (φA 𝐷)
4 elex 2542 . . . . . 6 (A 𝐷A V)
53, 4syl 14 . . . . 5 (φA V)
6 isset 2538 . . . . 5 (A V ↔ x x = A)
75, 6sylib 127 . . . 4 (φx x = A)
8 fvmptdv2.2 . . . . . 6 ((φ x = A) → B 𝑉)
9 elex 2542 . . . . . 6 (B 𝑉B V)
108, 9syl 14 . . . . 5 ((φ x = A) → B V)
112, 10eqeltrrd 2098 . . . 4 ((φ x = A) → 𝐶 V)
127, 11exlimddv 1762 . . 3 (φ𝐶 V)
131, 2, 3, 12fvmptd 5155 . 2 (φ → ((x 𝐷B)‘A) = 𝐶)
14 fveq1 5079 . . 3 (𝐹 = (x 𝐷B) → (𝐹A) = ((x 𝐷B)‘A))
1514eqeq1d 2031 . 2 (𝐹 = (x 𝐷B) → ((𝐹A) = 𝐶 ↔ ((x 𝐷B)‘A) = 𝐶))
1613, 15syl5ibrcom 146 1 (φ → (𝐹 = (x 𝐷B) → (𝐹A) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wex 1362   = wceq 1374   wcel 1376  Vcvv 2534   cmpt 3771  cfv 4806
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-14 1388  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005  ax-sep 3828  ax-pow 3880  ax-pr 3897
This theorem depends on definitions:  df-bi 110  df-3an 878  df-tru 1232  df-nf 1330  df-sb 1629  df-eu 1885  df-mo 1886  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-csb 2830  df-un 2901  df-in 2903  df-ss 2910  df-pw 3314  df-sn 3334  df-pr 3335  df-op 3337  df-uni 3534  df-br 3718  df-opab 3772  df-mpt 3773  df-id 3984  df-xp 4254  df-rel 4255  df-cnv 4256  df-co 4257  df-dm 4258  df-iota 4771  df-fun 4808  df-fv 4814
This theorem is referenced by: (None)
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