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Mirrors > Home > ILE Home > Th. List > fovcdmd | 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 6016 | . 2 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | |
5 | 1, 2, 3, 4 | syl3anc 1238 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 × cxp 4624 ⟶wf 5212 (class class class)co 5874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-ov 5877 |
This theorem is referenced by: eroveu 6625 isxmet2d 13741 ismet2 13747 comet 13892 bdmetval 13893 txmetcnp 13911 limccnp2lem 14038 limccnp2cntop 14039 |
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