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Mirrors > Home > ILE Home > Th. List > fovrn | GIF version |
Description: An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.) |
Ref | Expression |
---|---|
fovrn | ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4619 | . . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
2 | df-ov 5828 | . . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
3 | ffvelrn 5601 | . . . 4 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) → (𝐹‘〈𝐴, 𝐵〉) ∈ 𝐶) | |
4 | 2, 3 | eqeltrid 2244 | . . 3 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
5 | 1, 4 | sylan2 284 | . 2 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
6 | 5 | 3impb 1181 | 1 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 ∈ wcel 2128 〈cop 3563 × cxp 4585 ⟶wf 5167 ‘cfv 5171 (class class class)co 5825 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-ov 5828 |
This theorem is referenced by: fovrnda 5965 fovrnd 5966 ovmpoelrn 6156 mapxpen 6794 psmetcl 12768 xmetcl 12794 metcl 12795 blssm 12863 |
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