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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 5709 | . . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
2 | df-ov 7399 | . . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
3 | ffvelcdm 7071 | . . . 4 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) → (𝐹‘〈𝐴, 𝐵〉) ∈ 𝐶) | |
4 | 2, 3 | eqeltrid 2838 | . . 3 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
5 | 1, 4 | sylan2 594 | . 2 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
6 | 5 | 3impb 1116 | 1 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 〈cop 4630 × cxp 5670 ⟶wf 6531 ‘cfv 6535 (class class class)co 7396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-fv 6543 df-ov 7399 |
This theorem is referenced by: fovcdmda 7565 fovcdmd 7566 ovmpoelrn 8045 curry1f 8079 curry2f 8081 mapxpen 9131 axdc4lem 10437 axdc4uzlem 13935 imasmnd2 18649 grpsubcl 18890 imasgrp2 18925 imasring 20122 tsmsxplem1 23626 psmetcl 23782 xmetcl 23806 metcl 23807 blssm 23893 mbfi1fseqlem3 25204 mbfi1fseqlem4 25205 mbfi1fseqlem5 25206 grpocl 29718 grpodivcl 29757 vccl 29781 nvmcl 29864 cvmliftphtlem 34239 matunitlindflem1 36389 isbnd3 36558 clmgmOLD 36625 rngocl 36675 isdrngo2 36732 |
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