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Theorem fvmpt3 7020
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 2826 . . 3 (𝑥 = 𝐴 → (𝐵𝑉𝐶𝑉))
3 fvmpt3.c . . 3 (𝑥𝐷𝐵𝑉)
42, 3vtoclga 3577 . 2 (𝐴𝐷𝐶𝑉)
5 fvmpt3.b . . 3 𝐹 = (𝑥𝐷𝐵)
61, 5fvmptg 7014 . 2 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
74, 6mpdan 687 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cmpt 5225  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  fvmpt3i  7021  harval  9600  mrcfval  17651  elmptrab  23835  zringfrac  33582  frlmsnic  42550  wallispi  46085  1arymaptfv  48561  2arymaptfv  48572
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