Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fovrnd | 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 5921 | . 2 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | |
5 | 1, 2, 3, 4 | syl3anc 1217 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 × cxp 4545 ⟶wf 5127 (class class class)co 5782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 |
This theorem is referenced by: eroveu 6528 isxmet2d 12556 ismet2 12562 comet 12707 bdmetval 12708 txmetcnp 12726 limccnp2lem 12853 limccnp2cntop 12854 |
Copyright terms: Public domain | W3C validator |