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| Mirrors > Home > MPE Home > Th. List > fovcdm | Structured version Visualization version 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 5722 | . . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
| 2 | df-ov 7434 | . . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
| 3 | ffvelcdm 7101 | . . . 4 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) → (𝐹‘〈𝐴, 𝐵〉) ∈ 𝐶) | |
| 4 | 2, 3 | eqeltrid 2845 | . . 3 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
| 5 | 1, 4 | sylan2 593 | . 2 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
| 6 | 5 | 3impb 1115 | 1 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 〈cop 4632 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: fovcdmda 7604 fovcdmd 7605 ovmpoelrn 8097 curry1f 8131 curry2f 8133 mapxpen 9183 axdc4lem 10495 axdc4uzlem 14024 imasmnd2 18787 grpsubcl 19038 imasgrp2 19073 imasring 20327 tsmsxplem1 24161 psmetcl 24317 xmetcl 24341 metcl 24342 blssm 24428 mbfi1fseqlem3 25752 mbfi1fseqlem4 25753 mbfi1fseqlem5 25754 grpocl 30519 grpodivcl 30558 vccl 30582 nvmcl 30665 cvmliftphtlem 35322 matunitlindflem1 37623 isbnd3 37791 clmgmOLD 37858 rngocl 37908 isdrngo2 37965 |
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