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Theorem mapfvd 8820
Description: The value of a function that maps from 𝐵 to 𝐴. (Contributed by AV, 2-Feb-2023.)
Hypotheses
Ref Expression
mapfvd.m 𝑀 = (𝐴m 𝐵)
mapfvd.f (𝜑𝐹𝑀)
mapfvd.x (𝜑𝑋𝐵)
Assertion
Ref Expression
mapfvd (𝜑 → (𝐹𝑋) ∈ 𝐴)

Proof of Theorem mapfvd
StepHypRef Expression
1 mapfvd.f . 2 (𝜑𝐹𝑀)
2 mapfvd.x . . . 4 (𝜑𝑋𝐵)
3 elmapi 8790 . . . 4 (𝐹 ∈ (𝐴m 𝐵) → 𝐹:𝐵𝐴)
4 ffvelcdm 7033 . . . . 5 ((𝐹:𝐵𝐴𝑋𝐵) → (𝐹𝑋) ∈ 𝐴)
54expcom 415 . . . 4 (𝑋𝐵 → (𝐹:𝐵𝐴 → (𝐹𝑋) ∈ 𝐴))
62, 3, 5syl2imc 41 . . 3 (𝐹 ∈ (𝐴m 𝐵) → (𝜑 → (𝐹𝑋) ∈ 𝐴))
7 mapfvd.m . . 3 𝑀 = (𝐴m 𝐵)
86, 7eleq2s 2852 . 2 (𝐹𝑀 → (𝜑 → (𝐹𝑋) ∈ 𝐴))
91, 8mpcom 38 1 (𝜑 → (𝐹𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wf 6493  cfv 6497  (class class class)co 7358  m cmap 8768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770
This theorem is referenced by:  fsuppind  40808  1arympt1  46810  rrx2pxel  46883  rrx2pyel  46884
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