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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 6596 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {⟨𝐴, 𝐵⟩}) | |
2 | opex 5463 | . . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V | |
3 | 2 | snid 4663 | . 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} |
4 | funopfv 6940 | . 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 1541 ∈ wcel 2106 {csn 4627 ⟨cop 4633 Fun wfun 6534 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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: fvsn 7175 fvsnun1 7176 fsnunfv 7181 fvpr1g 7184 fvpr2gOLD 7186 fsnex 7277 suppsnop 8159 mapsnend 9032 enfixsn 9077 axdc3lem4 10444 fseq1p1m1 13571 1fv 13616 s1fv 14556 sumsnf 15685 prodsn 15902 prodsnf 15904 seq1st 16504 vdwlem8 16917 setsid 17137 mgm1 18573 sgrp1 18616 mnd1 18663 mnd1id 18664 gsumws1 18715 grp1 18926 dprdsn 19900 ring1 20115 ixpsnbasval 20824 frgpcyg 21120 mat1dimscm 21968 mat1dimmul 21969 mat1rhmelval 21973 m1detdiag 22090 pt1hmeo 23301 noextenddif 27160 noextendlt 27161 noextendgt 27162 1loopgrvd0 28750 1hevtxdg0 28751 1hevtxdg1 28752 1egrvtxdg1 28755 wlkl0 29609 actfunsnrndisj 33605 reprsuc 33615 breprexplema 33630 cvmliftlem7 34270 cvmliftlem13 34275 bj-fununsn2 36123 sticksstones9 40958 sticksstones11 40960 metakunt20 40992 frlmsnic 41107 sumsnd 43695 ovnovollem1 45358 nnsum3primesprm 46444 lincvalsng 47050 snlindsntorlem 47104 lmod1lem2 47122 lmod1lem3 47123 0aryfvalelfv 47274 1arympt1fv 47278 |
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