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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.) (Proof shortened by BJ, 25-Feb-2023.) |
Ref | Expression |
---|---|
fvsn.1 | ⊢ 𝐴 ∈ V |
fvsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fvsn | ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | fvsn.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | fvsng 7200 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 |
This theorem is referenced by: frrlem12 8321 elixpsn 8976 ac6sfi 9318 dcomex 10485 axdc3lem4 10491 0ram 17054 mdet0fv0 22616 chpmat0d 22856 imasdsf1olem 24399 axlowdimlem8 28979 axlowdimlem11 28982 subfacp1lem2a 35165 subfacp1lem5 35169 cvmliftlem4 35273 finixpnum 37592 poimirlem3 37610 fdc 37732 grposnOLD 37869 1arymaptfo 48493 mndtchom 48893 mndtcco 48894 |
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