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Theorem fvmptdv2 5169
Description: Alternate deduction version of fvmpt 5158, 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 2010 . . 3 (φ → (x 𝐷B) = (x 𝐷B))
2 fvmptdv2.3 . . 3 ((φ x = A) → B = 𝐶)
3 fvmptdv2.1 . . 3 (φA 𝐷)
4 elex 2530 . . . . . 6 (A 𝐷A V)
53, 4syl 14 . . . . 5 (φA V)
6 isset 2526 . . . . 5 (A V ↔ x x = A)
75, 6sylib 127 . . . 4 (φx x = A)
8 fvmptdv2.2 . . . . . 6 ((φ x = A) → B 𝑉)
9 elex 2530 . . . . . 6 (B 𝑉B V)
108, 9syl 14 . . . . 5 ((φ x = A) → B V)
112, 10eqeltrrd 2084 . . . 4 ((φ x = A) → 𝐶 V)
127, 11exlimddv 1747 . . 3 (φ𝐶 V)
131, 2, 3, 12fvmptd 5162 . 2 (φ → ((x 𝐷B)‘A) = 𝐶)
14 fveq1 5086 . . 3 (𝐹 = (x 𝐷B) → (𝐹A) = ((x 𝐷B)‘A))
1514eqeq1d 2017 . 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   = wceq 1219  wex 1350   wcel 1362  Vcvv 2522  cmpt 3777  cfv 4813
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 613  ax-5 1305  ax-7 1306  ax-gen 1307  ax-ie1 1351  ax-ie2 1352  ax-8 1364  ax-10 1365  ax-11 1366  ax-i12 1367  ax-bnd 1368  ax-4 1369  ax-14 1374  ax-17 1388  ax-i9 1392  ax-ial 1396  ax-i5r 1397  ax-ext 1991  ax-sep 3834  ax-pow 3886  ax-pr 3903
This theorem depends on definitions:  df-bi 110  df-3an 868  df-tru 1222  df-nf 1319  df-sb 1615  df-eu 1872  df-mo 1873  df-clab 1996  df-cleq 2002  df-clel 2005  df-nfc 2136  df-ral 2276  df-rex 2277  df-v 2524  df-sbc 2729  df-csb 2817  df-un 2886  df-in 2888  df-ss 2895  df-pw 3321  df-sn 3341  df-pr 3342  df-op 3344  df-uni 3540  df-br 3724  df-opab 3778  df-mpt 3779  df-id 3989  df-xp 4262  df-rel 4263  df-cnv 4264  df-co 4265  df-dm 4266  df-iota 4778  df-fun 4815  df-fv 4821
This theorem is referenced by: (None)
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