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| Mirrors > Home > MPE Home > Th. List > fovcdmd | 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) |
| fovcdmd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑅) |
| fovcdmd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
| Ref | Expression |
|---|---|
| fovcdmd | ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fovcdmd.1 | . 2 ⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) | |
| 2 | fovcdmd.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑅) | |
| 3 | fovcdmd.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
| 4 | fovcdm 7603 | . 2 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 × cxp 5683 ⟶wf 6557 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: eroveu 8852 fseqenlem1 10064 rlimcn2 15627 homarel 18081 curf1cl 18273 curf2cl 18276 hofcllem 18303 yonedalem3b 18324 gasubg 19320 gacan 19323 gapm 19324 gastacos 19328 orbsta 19331 galactghm 19422 sylow1lem2 19617 sylow2alem2 19636 sylow3lem1 19645 efgcpbllemb 19773 frgpuplem 19790 frlmbas3 21796 mamucl 22405 mamuass 22406 mamudi 22407 mamudir 22408 mamuvs1 22409 mamuvs2 22410 mamulid 22447 mamurid 22448 mamutpos 22464 matgsumcl 22466 mavmulcl 22553 mavmulass 22555 mdetleib2 22594 mdetf 22601 mdetdiaglem 22604 mdetrlin 22608 mdetrsca 22609 mdetralt 22614 mdetunilem7 22624 maducoeval2 22646 madugsum 22649 madurid 22650 tsmsxplem2 24162 isxmet2d 24337 ismet2 24343 prdsxmetlem 24378 comet 24526 ipcn 25280 ovoliunlem2 25538 itg1addlem4 25734 itg1addlem5 25735 mbfi1fseqlem5 25754 limccnp2 25927 midcl 28785 fedgmullem2 33681 pstmxmet 33896 cvmlift2lem9 35316 isbnd3 37791 prdsbnd 37800 iscringd 38005 rmxycomplete 42929 rmxyadd 42933 2arympt 48570 |
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