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Theorem fvmpt2d 5507
 Description: Deduction version of fvmpt2 5504. (Contributed by Thierry Arnoux, 8-Dec-2016.)
Hypotheses
Ref Expression
fvmpt2d.1 (𝜑𝐹 = (𝑥𝐴𝐵))
fvmpt2d.4 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
fvmpt2d ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmpt2d
StepHypRef Expression
1 fvmpt2d.1 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
21fveq1d 5423 . . 3 (𝜑 → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
32adantr 274 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4 simpr 109 . . 3 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 fvmpt2d.4 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
6 eqid 2139 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76fvmpt2 5504 . . 3 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
84, 5, 7syl2anc 408 . 2 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
93, 8eqtrd 2172 1 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ↦ cmpt 3989  ‘cfv 5123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131 This theorem is referenced by:  iseqf1olemjpcl  10275  iseqf1olemqpcl  10276  isumshft  11266
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