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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 7173 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4627 〈cop 4633 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 |
This theorem is referenced by: fvpr1OLD 7187 frrlem12 8277 elixpsn 8927 ac6sfi 9283 dcomex 10438 axdc3lem4 10444 0ram 16949 mdet0fv0 22078 chpmat0d 22318 imasdsf1olem 23861 axlowdimlem8 28187 axlowdimlem11 28190 subfacp1lem2a 34109 subfacp1lem5 34113 cvmliftlem4 34217 finixpnum 36411 poimirlem3 36429 fdc 36551 grposnOLD 36688 1arymaptfo 47231 mndtchom 47612 mndtcco 47613 |
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