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Theorem bj-mptval 37118
Description: Value of a function given in maps-to notation. (Contributed by BJ, 30-Dec-2020.)
Hypothesis
Ref Expression
bj-mptval.nf 𝑥𝐴
Assertion
Ref Expression
bj-mptval (∀𝑥𝐴 𝐵𝑉 → (𝑋𝐴 → (((𝑥𝐴𝐵)‘𝑋) = 𝑌𝑋(𝑥𝐴𝐵)𝑌)))

Proof of Theorem bj-mptval
StepHypRef Expression
1 bj-mptval.nf . . 3 𝑥𝐴
21fnmptf 6704 . 2 (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)
3 fnbrfvb 6959 . . 3 (((𝑥𝐴𝐵) Fn 𝐴𝑋𝐴) → (((𝑥𝐴𝐵)‘𝑋) = 𝑌𝑋(𝑥𝐴𝐵)𝑌))
43ex 412 . 2 ((𝑥𝐴𝐵) Fn 𝐴 → (𝑋𝐴 → (((𝑥𝐴𝐵)‘𝑋) = 𝑌𝑋(𝑥𝐴𝐵)𝑌)))
52, 4syl 17 1 (∀𝑥𝐴 𝐵𝑉 → (𝑋𝐴 → (((𝑥𝐴𝐵)‘𝑋) = 𝑌𝑋(𝑥𝐴𝐵)𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wnfc 2890  wral 3061   class class class wbr 5143  cmpt 5225   Fn wfn 6556  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-ne 2941  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-fn 6564  df-fv 6569
This theorem is referenced by: (None)
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