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Mirrors > Home > ILE Home > Th. List > fvsn | GIF version |
Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.) |
Ref | Expression |
---|---|
fvsn.1 | ⊢ 𝐴 ∈ V |
fvsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fvsn | ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | fvsn.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | funsn 5179 | . 2 ⊢ Fun {〈𝐴, 𝐵〉} |
4 | 1, 2 | opex 4159 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V |
5 | 4 | snid 3563 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} |
6 | funopfv 5469 | . 2 ⊢ (Fun {〈𝐴, 𝐵〉} → (〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵)) | |
7 | 3, 5, 6 | mp2 16 | 1 ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 Vcvv 2689 {csn 3532 〈cop 3535 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: fvsng 5624 fvsnun1 5625 fvpr1 5632 elixpsn 6637 mapsnen 6713 ac6sfi 6800 |
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