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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 8411 | . . . 4 ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) → 𝐹:𝐵⟶𝐴) | |
4 | ffvelrn 6826 | . . . . 5 ⊢ ((𝐹:𝐵⟶𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝐴) | |
5 | 4 | expcom 417 | . . . 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 2908 | . 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 1538 ∈ wcel 2111 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 |
This theorem is referenced by: fsuppind 39456 1arympt1 45052 rrx2pxel 45125 rrx2pyel 45126 |
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