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Theorem mapfvd 8421
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 8406 . . . 4 (𝐹 ∈ (𝐴m 𝐵) → 𝐹:𝐵𝐴)
4 ffvelrn 6825 . . . . 5 ((𝐹:𝐵𝐴𝑋𝐵) → (𝐹𝑋) ∈ 𝐴)
54expcom 416 . . . 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 2929 . 2 (𝐹𝑀 → (𝜑 → (𝐹𝑋) ∈ 𝐴))
91, 8mpcom 38 1 (𝜑 → (𝐹𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  wf 6327  cfv 6331  (class class class)co 7133  m cmap 8384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-map 8386
This theorem is referenced by:  rrx2pxel  44885  rrx2pyel  44886
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