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Theorem fvmpt3 6949
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 2822 . . 3 (𝑥 = 𝐴 → (𝐵𝑉𝐶𝑉))
3 fvmpt3.c . . 3 (𝑥𝐷𝐵𝑉)
42, 3vtoclga 3532 . 2 (𝐴𝐷𝐶𝑉)
5 fvmpt3.b . . 3 𝐹 = (𝑥𝐷𝐵)
61, 5fvmptg 6943 . 2 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
74, 6mpdan 685 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cmpt 5186  cfv 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501
This theorem is referenced by:  fvmpt3i  6950  harval  9492  mrcfval  17480  elmptrab  23162  frlmsnic  40689  wallispi  44243  1arymaptfv  46658  2arymaptfv  46669
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