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Mirrors > Home > MPE Home > Th. List > fvsng | Structured version Visualization version GIF version |
Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.) (Proof shortened by BJ, 25-Feb-2023.) |
Ref | Expression |
---|---|
fvsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funsng 6398 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) | |
2 | opex 5347 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | snid 4591 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} |
4 | funopfv 6710 | . 2 ⊢ (Fun {〈𝐴, 𝐵〉} → (〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵)) | |
5 | 1, 3, 4 | mpisyl 21 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 〈cop 4563 Fun wfun 6342 ‘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: fvsn 6935 fvsnun1 6936 fsnunfv 6941 fvpr1g 6946 fvpr2g 6947 fsnex 7030 suppsnop 7833 mapsnend 8576 enfixsn 8614 axdc3lem4 9863 fseq1p1m1 12969 1fv 13014 s1fv 13952 sumsnf 15087 prodsn 15304 prodsnf 15306 seq1st 15903 vdwlem8 16312 setsid 16526 mgm1 17856 sgrp1 17898 mnd1 17940 mnd1id 17941 gsumws1 17990 grp1 18144 dprdsn 19087 ring1 19281 ixpsnbasval 19910 frgpcyg 20648 mat1dimscm 21012 mat1dimmul 21013 mat1rhmelval 21017 m1detdiag 21134 pt1hmeo 22342 1loopgrvd0 27213 1hevtxdg0 27214 1hevtxdg1 27215 1egrvtxdg1 27218 wlkl0 28073 actfunsnrndisj 31775 reprsuc 31785 breprexplema 31800 cvmliftlem7 32435 cvmliftlem13 32440 noextenddif 33072 noextendlt 33073 noextendgt 33074 bj-fununsn2 34428 frlmsnic 39027 sumsnd 41160 ovnovollem1 42815 nnsum3primesprm 43832 lincvalsng 44399 snlindsntorlem 44453 lmod1lem2 44471 lmod1lem3 44472 |
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