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Theorem mapfvd 8898
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 8868 . . . 4 (𝐹 ∈ (𝐴m 𝐵) → 𝐹:𝐵𝐴)
4 ffvelcdm 7091 . . . . 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 2847 . 2 (𝐹𝑀 → (𝜑 → (𝐹𝑋) ∈ 𝐴))
91, 8mpcom 38 1 (𝜑 → (𝐹𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wf 6544  cfv 6548  (class class class)co 7420  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-map 8847
This theorem is referenced by:  fsuppind  41823  1arympt1  47711  rrx2pxel  47784  rrx2pyel  47785
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