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Mirrors > Home > ILE Home > Th. List > funopfv | 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 4019 | . 2 ⊢ (𝐴𝐹𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹) | |
2 | funbrfv 5575 | . 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵)) | |
3 | 1, 2 | biimtrrid 153 | 1 ⊢ (Fun 𝐹 → (〈𝐴, 𝐵〉 ∈ 𝐹 → (𝐹‘𝐴) = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 〈cop 3610 class class class wbr 4018 Fun wfun 5229 ‘cfv 5235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 |
This theorem is referenced by: fvopab3ig 5611 fvsn 5732 ovidig 6015 ovigg 6018 f1o2ndf1 6254 fundmen 6833 frecuzrdg0 10446 frecuzrdgsuc 10447 frecuzrdg0t 10455 frecuzrdgsuctlem 10456 strslfvd 12557 strslfv2d 12558 imasaddvallemg 12795 |
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