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Theorem fvmpt2d 5200
Description: Deduction version of fvmpt2 5197. (Contributed by Thierry Arnoux, 8-Dec-2016.)
Hypotheses
Ref Expression
fvmpt2d.1 (φ𝐹 = (x AB))
fvmpt2d.4 ((φ x A) → B 𝑉)
Assertion
Ref Expression
fvmpt2d ((φ x A) → (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 = (x AB))
21fveq1d 5123 . . 3 (φ → (𝐹x) = ((x AB)‘x))
32adantr 261 . 2 ((φ x A) → (𝐹x) = ((x AB)‘x))
4 simpr 103 . . 3 ((φ x A) → x A)
5 fvmpt2d.4 . . 3 ((φ x A) → B 𝑉)
6 eqid 2037 . . . 4 (x AB) = (x AB)
76fvmpt2 5197 . . 3 ((x A B 𝑉) → ((x AB)‘x) = B)
84, 5, 7syl2anc 391 . 2 ((φ x A) → ((x AB)‘x) = B)
93, 8eqtrd 2069 1 ((φ x A) → (𝐹x) = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cmpt 3809  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-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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by: (None)
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