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Theorem mapfvd 8870
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 8840 . . . 4 (𝐹 ∈ (𝐴m 𝐵) → 𝐹:𝐵𝐴)
4 ffvelcdm 7074 . . . . 5 ((𝐹:𝐵𝐴𝑋𝐵) → (𝐹𝑋) ∈ 𝐴)
54expcom 413 . . . 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 2843 . 2 (𝐹𝑀 → (𝜑 → (𝐹𝑋) ∈ 𝐴))
91, 8mpcom 38 1 (𝜑 → (𝐹𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wf 6530  cfv 6534  (class class class)co 7402  m cmap 8817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-map 8819
This theorem is referenced by:  fsuppind  41693  1arympt1  47573  rrx2pxel  47646  rrx2pyel  47647
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