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Mirrors > Home > MPE Home > Th. List > fovcdmda | Structured version Visualization version GIF version |
Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
Ref | Expression |
---|---|
fovcdmd.1 | ⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) |
Ref | Expression |
---|---|
fovcdmda | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fovcdmd.1 | . . 3 ⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) | |
2 | fovcdm 7602 | . . 3 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | |
3 | 1, 2 | syl3an1 1162 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
4 | 3 | 3expb 1119 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 × cxp 5686 ⟶wf 6558 (class class class)co 7430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 |
This theorem is referenced by: eroprf 8853 yonedalem3 18336 yonedainv 18337 gass 19331 gsumxp2 20012 mamulid 22462 mamurid 22463 maducoeval2 22661 madutpos 22663 madugsum 22664 madurid 22665 isxmet2d 24352 prdsxmetlem 24393 rrxds 25440 ofrn 32655 fedgmullem2 33657 metideq 33853 sibfof 34321 ofoacl 43346 naddcnfcl 43354 |
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