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| Mirrors > Home > MPE Home > Th. List > funopfv | Structured version Visualization version GIF version | ||
| Description: The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.) |
| Ref | Expression |
|---|---|
| funopfv | ⊢ (Fun 𝐹 → (〈𝐴, 𝐵〉 ∈ 𝐹 → (𝐹‘𝐴) = 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-br 5144 | . 2 ⊢ (𝐴𝐹𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹) | |
| 2 | funbrfv 6957 | . 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵)) | |
| 3 | 1, 2 | biimtrrid 243 | 1 ⊢ (Fun 𝐹 → (〈𝐴, 𝐵〉 ∈ 𝐹 → (𝐹‘𝐴) = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 Fun wfun 6555 ‘cfv 6561 |
| 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-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-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: fvopab3ig 7012 fvsng 7200 fveqf1o 7322 ovidig 7575 ovigg 7578 funfv1st2nd 8071 funelss 8072 f1o2ndf1 8147 fundmen 9071 dif1en 9200 dif1enOLD 9202 uzrdg0i 14000 uzrdgsuci 14001 strfvd 17237 strfv2d 17238 imasaddvallem 17574 imasvscafn 17582 noseqrdg0 28313 noseqrdgsuc 28314 adjeq 31954 bnj1379 34844 bnj97 34880 bnj553 34912 bnj966 34958 bnj1442 35063 satfv0fvfmla0 35418 satfv1fvfmla1 35428 |
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