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Mirrors > Home > MPE Home > Th. List > fovrnd | Structured version Visualization version GIF version |
Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
Ref | Expression |
---|---|
fovrnd.1 | ⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) |
fovrnd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑅) |
fovrnd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
Ref | Expression |
---|---|
fovrnd | ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fovrnd.1 | . 2 ⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) | |
2 | fovrnd.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑅) | |
3 | fovrnd.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
4 | fovrn 7307 | . 2 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | |
5 | 1, 2, 3, 4 | syl3anc 1363 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 × cxp 5546 ⟶wf 6344 (class class class)co 7145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 |
This theorem is referenced by: eroveu 8381 fseqenlem1 9438 rlimcn2 14935 homarel 17284 curf1cl 17466 curf2cl 17469 hofcllem 17496 yonedalem3b 17517 gasubg 18370 gacan 18373 gapm 18374 gastacos 18378 orbsta 18381 galactghm 18461 sylow1lem2 18653 sylow2alem2 18672 sylow3lem1 18681 efgcpbllemb 18810 frgpuplem 18827 frlmbas3 20848 mamucl 20938 mamuass 20939 mamudi 20940 mamudir 20941 mamuvs1 20942 mamuvs2 20943 mamulid 20978 mamurid 20979 mamutpos 20995 matgsumcl 20997 mavmulcl 21084 mavmulass 21086 mdetleib2 21125 mdetf 21132 mdetdiaglem 21135 mdetrlin 21139 mdetrsca 21140 mdetralt 21145 mdetunilem7 21155 maducoeval2 21177 madugsum 21180 madurid 21181 tsmsxplem2 22689 isxmet2d 22864 ismet2 22870 prdsxmetlem 22905 comet 23050 ipcn 23776 ovoliunlem2 24031 itg1addlem4 24227 itg1addlem5 24228 mbfi1fseqlem5 24247 limccnp2 24417 midcl 26490 fedgmullem2 30925 pstmxmet 31036 cvmlift2lem9 32455 isbnd3 34943 prdsbnd 34952 iscringd 35157 rmxycomplete 39392 rmxyadd 39396 |
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