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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 5592 | . . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
2 | df-ov 7159 | . . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
3 | ffvelrn 6849 | . . . 4 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) → (𝐹‘〈𝐴, 𝐵〉) ∈ 𝐶) | |
4 | 2, 3 | eqeltrid 2917 | . . 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 1111 | 1 ⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 〈cop 4573 × cxp 5553 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 |
This theorem is referenced by: fovrnda 7319 fovrnd 7320 ovmpoelrn 7770 curry1f 7801 curry2f 7803 mapxpen 8683 axdc4lem 9877 axdc4uzlem 13352 imasmnd2 17948 grpsubcl 18179 imasgrp2 18214 imasring 19369 tsmsxplem1 22761 psmetcl 22917 xmetcl 22941 metcl 22942 blssm 23028 mbfi1fseqlem3 24318 mbfi1fseqlem4 24319 mbfi1fseqlem5 24320 grpocl 28277 grpodivcl 28316 vccl 28340 nvmcl 28423 cvmliftphtlem 32564 matunitlindflem1 34903 isbnd3 35077 clmgmOLD 35144 rngocl 35194 isdrngo2 35251 |
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