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Mirrors > Home > MPE Home > Th. List > mapfvd | Structured version Visualization version GIF version |
Description: The value of a function that maps from 𝐵 to 𝐴. (Contributed by AV, 2-Feb-2023.) |
Ref | Expression |
---|---|
mapfvd.m | ⊢ 𝑀 = (𝐴 ↑𝑚 𝐵) |
mapfvd.f | ⊢ (𝜑 → 𝐹 ∈ 𝑀) |
mapfvd.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
mapfvd | ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴) |
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) |
Colors of variables: wff setvar class |
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 |
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