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Theorem fvmptdv2 5206
Description: Alternate deduction version of fvmpt 5195, 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 2041 . . 3 (φ → (x 𝐷B) = (x 𝐷B))
2 fvmptdv2.3 . . 3 ((φ x = A) → B = 𝐶)
3 fvmptdv2.1 . . 3 (φA 𝐷)
4 elex 2563 . . . . . 6 (A 𝐷A V)
53, 4syl 14 . . . . 5 (φA V)
6 isset 2558 . . . . 5 (A V ↔ x x = A)
75, 6sylib 127 . . . 4 (φx x = A)
8 fvmptdv2.2 . . . . . 6 ((φ x = A) → B 𝑉)
9 elex 2563 . . . . . 6 (B 𝑉B V)
108, 9syl 14 . . . . 5 ((φ x = A) → B V)
112, 10eqeltrrd 2115 . . . 4 ((φ x = A) → 𝐶 V)
127, 11exlimddv 1778 . . 3 (φ𝐶 V)
131, 2, 3, 12fvmptd 5199 . 2 (φ → ((x 𝐷B)‘A) = 𝐶)
14 fveq1 5123 . . 3 (𝐹 = (x 𝐷B) → (𝐹A) = ((x 𝐷B)‘A))
1514eqeq1d 2048 . 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 1243  wex 1381   wcel 1393  Vcvv 2554  cmpt 3812  cfv 4848
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3869  ax-pow 3921  ax-pr 3938
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-csb 2850  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-uni 3575  df-br 3759  df-opab 3813  df-mpt 3814  df-id 4024  df-xp 4297  df-rel 4298  df-cnv 4299  df-co 4300  df-dm 4301  df-iota 4813  df-fun 4850  df-fv 4856
This theorem is referenced by: (None)
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