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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 6934 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 |
This theorem is referenced by: fvpr1 6944 elixpsn 8489 ac6sfi 8750 dcomex 9857 axdc3lem4 9863 0ram 16344 mdet0fv0 21131 chpmat0d 21370 imasdsf1olem 22910 axlowdimlem8 26662 axlowdimlem11 26665 subfacp1lem2a 32324 subfacp1lem5 32328 cvmliftlem4 32432 frrlem12 33031 finixpnum 34758 poimirlem3 34776 fdc 34901 grposnOLD 35041 |
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