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Mirrors > Home > ILE Home > Th. List > fovcdm | GIF version |
Description: An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.) |
Ref | Expression |
---|---|
fovcdm | ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4660 | . . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)) | |
2 | df-ov 5880 | . . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩) | |
3 | ffvelcdm 5651 | . . . 4 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶) | |
4 | 2, 3 | eqeltrid 2264 | . . 3 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
5 | 1, 4 | sylan2 286 | . 2 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
6 | 5 | 3impb 1199 | 1 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 ∈ wcel 2148 ⟨cop 3597 × cxp 4626 ⟶wf 5214 ‘cfv 5218 (class class class)co 5877 |
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 4123 ax-pow 4176 ax-pr 4211 |
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 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5880 |
This theorem is referenced by: fovcdmda 6020 fovcdmd 6021 ovmpoelrn 6210 mapxpen 6850 grpsubcl 12955 psmetcl 13911 xmetcl 13937 metcl 13938 blssm 14006 |
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