Theorem mapfvd 8164

Description: The value of a function that maps from 𝐵 to 𝐴. (Contributed by AV, 2-Feb-2023.)

Hypotheses

Ref	Expression
mapfvd.m	⊢ 𝑀 = (𝐴 ↑𝑚 𝐵)
mapfvd.f	⊢ (𝜑 → 𝐹 ∈ 𝑀)
mapfvd.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
mapfvd	⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴)

Proof of Theorem mapfvd

Step	Hyp	Ref	Expression
1		mapfvd.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑀)
2		mapfvd.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		elmapi 8149	. . . 4 ⊢ (𝐹 ∈ (𝐴 ↑𝑚 𝐵) → 𝐹:𝐵⟶𝐴)
4		ffvelrn 6611	. . . . 5 ⊢ ((𝐹:𝐵⟶𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝐴)
5	4	expcom 404	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (𝐹:𝐵⟶𝐴 → (𝐹‘𝑋) ∈ 𝐴))
6	2, 3, 5	syl2imc 41	. . 3 ⊢ (𝐹 ∈ (𝐴 ↑𝑚 𝐵) → (𝜑 → (𝐹‘𝑋) ∈ 𝐴))
7		mapfvd.m	. . 3 ⊢ 𝑀 = (𝐴 ↑𝑚 𝐵)
8	6, 7	eleq2s 2924	. 2 ⊢ (𝐹 ∈ 𝑀 → (𝜑 → (𝐹‘𝑋) ∈ 𝐴))
9	1, 8	mpcom 38	1 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910   ↑_𝑚 cmap 8127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129

This theorem is referenced by:  rrx2pxel  42271  rrx2pyel  42272