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Mirrors > Home > MPE Home > Th. List > fovrnda | 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) |
Ref | Expression |
---|---|
fovrnda | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fovrnd.1 | . . 3 ⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) | |
2 | fovrn 7307 | . . 3 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | |
3 | 1, 2 | syl3an1 1155 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
4 | 3 | 3expb 1112 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ 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: eroprf 8384 yonedalem3 17518 yonedainv 17519 gass 18369 gsumxp2 19029 mamulid 20978 mamurid 20979 maducoeval2 21177 madutpos 21179 madugsum 21180 madurid 21181 isxmet2d 22864 prdsxmetlem 22905 rrxds 23923 ofrn 30314 fedgmullem2 30925 metideq 31032 sibfof 31497 |
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