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Theorem fvmpt3 6325
Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
fvmpt3.a (𝑥 = 𝐴𝐵 = 𝐶)
fvmpt3.b 𝐹 = (𝑥𝐷𝐵)
fvmpt3.c (𝑥𝐷𝐵𝑉)
Assertion
Ref Expression
fvmpt3 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑉
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt3
StepHypRef Expression
1 fvmpt3.a . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
21eleq1d 2715 . . 3 (𝑥 = 𝐴 → (𝐵𝑉𝐶𝑉))
3 fvmpt3.c . . 3 (𝑥𝐷𝐵𝑉)
42, 3vtoclga 3303 . 2 (𝐴𝐷𝐶𝑉)
5 fvmpt3.b . . 3 𝐹 = (𝑥𝐷𝐵)
61, 5fvmptg 6319 . 2 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
74, 6mpdan 703 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cmpt 4762  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  fvmpt3i  6326  harval  8508  mrcfval  16315  elmptrab  21678  wallispi  40605
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