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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 3938 | . 2 ⊢ (𝐴𝐹𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹) | |
2 | funbrfv 5468 | . 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵)) | |
3 | 1, 2 | syl5bir 152 | 1 ⊢ (Fun 𝐹 → (〈𝐴, 𝐵〉 ∈ 𝐹 → (𝐹‘𝐴) = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 〈cop 3535 class class class wbr 3937 Fun wfun 5125 ‘cfv 5131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 |
This theorem is referenced by: fvopab3ig 5503 fvsn 5623 ovidig 5896 ovigg 5899 f1o2ndf1 6133 fundmen 6708 frecuzrdg0 10217 frecuzrdgsuc 10218 frecuzrdg0t 10226 frecuzrdgsuctlem 10227 strslfvd 12039 strslfv2d 12040 |
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