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Mirrors > Home > MPE Home > Th. List > fovrn | Structured version Visualization version 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 5305 | . . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
2 | df-ov 6816 | . . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
3 | ffvelrn 6520 | . . . 4 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) → (𝐹‘〈𝐴, 𝐵〉) ∈ 𝐶) | |
4 | 2, 3 | syl5eqel 2843 | . . 3 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
5 | 1, 4 | sylan2 492 | . 2 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) |
6 | 5 | 3impb 1108 | 1 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 ∈ wcel 2139 〈cop 4327 × cxp 5264 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-fv 6057 df-ov 6816 |
This theorem is referenced by: fovrnda 6970 fovrnd 6971 ovmpt2elrn 7409 curry1f 7439 curry2f 7441 mapxpen 8291 axdc4lem 9469 axdc4uzlem 12976 imasmnd2 17528 grpsubcl 17696 imasgrp2 17731 imasring 18819 tsmsxplem1 22157 psmetcl 22313 xmetcl 22337 metcl 22338 blssm 22424 mbfi1fseqlem3 23683 mbfi1fseqlem4 23684 mbfi1fseqlem5 23685 grpocl 27663 grpodivcl 27702 vccl 27727 nvmcl 27810 cvmliftphtlem 31606 matunitlindflem1 33718 isbnd3 33896 clmgmOLD 33963 rngocl 34013 isdrngo2 34070 |
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