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Mirrors > Home > MPE Home > Th. List > fvsn | Structured version Visualization version 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 6100 | . 2 ⊢ Fun {〈𝐴, 𝐵〉} |
4 | opex 5081 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
5 | 4 | snid 4353 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} |
6 | funopfv 6396 | . 2 ⊢ (Fun {〈𝐴, 𝐵〉} → (〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵)) | |
7 | 3, 5, 6 | mp2 9 | 1 ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 Vcvv 3340 {csn 4321 〈cop 4327 Fun wfun 6043 ‘cfv 6049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-iota 6012 df-fun 6051 df-fv 6057 |
This theorem is referenced by: fvsng 6611 fvsnun1 6612 fvpr1 6620 elixpsn 8113 mapsnen 8200 ac6sfi 8369 dcomex 9461 axdc3lem4 9467 0ram 15926 mdet0fv0 20602 chpmat0d 20841 imasdsf1olem 22379 axlowdimlem8 26028 axlowdimlem11 26031 subfacp1lem2a 31469 subfacp1lem5 31473 cvmliftlem4 31577 finixpnum 33707 poimirlem3 33725 fdc 33854 grposnOLD 33994 |
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