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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 4686 | . 2 ⊢ (𝐴𝐹𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹) | |
2 | funbrfv 6272 | . 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵)) | |
3 | 1, 2 | syl5bir 233 | 1 ⊢ (Fun 𝐹 → (〈𝐴, 𝐵〉 ∈ 𝐹 → (𝐹‘𝐴) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 〈cop 4216 class class class wbr 4685 Fun wfun 5920 ‘cfv 5926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 |
This theorem is referenced by: fvopab3ig 6317 fvsn 6487 fveqf1o 6597 ovidig 6820 ovigg 6823 f1o2ndf1 7330 fundmen 8071 uzrdg0i 12798 uzrdgsuci 12799 strfvd 15951 strfv2d 15952 imasaddvallem 16236 imasvscafn 16244 basvtxvalOLD 25948 edgfiedgvalOLD 25949 adjeq 28922 bnj1379 31027 bnj97 31062 bnj553 31094 bnj966 31140 bnj1442 31243 |
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