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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 8870 | . . . 4 ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) → 𝐹:𝐵⟶𝐴) | |
4 | ffvelcdm 7087 | . . . . 5 ⊢ ((𝐹:𝐵⟶𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝐴) | |
5 | 4 | expcom 412 | . . . 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 2844 | . 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 1534 ∈ wcel 2099 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 ↑m cmap 8847 |
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 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-map 8849 |
This theorem is referenced by: fsuppind 42280 1arympt1 48062 rrx2pxel 48135 rrx2pyel 48136 |
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