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| 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 6617 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) | |
| 2 | opex 5469 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 3 | 2 | snid 4662 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} | 
| 4 | funopfv 6958 | . 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 395 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 Fun wfun 6555 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 | 
| This theorem is referenced by: fvsn 7201 fvsnun1 7202 fsnunfv 7207 fvpr1g 7210 fsnex 7303 suppsnop 8203 mapsnend 9076 enfixsn 9121 axdc3lem4 10493 fseq1p1m1 13638 1fv 13687 s1fv 14648 sumsnf 15779 prodsn 15998 prodsnf 16000 seq1st 16608 vdwlem8 17026 setsid 17244 mgm1 18671 sgrp1 18742 mnd1 18792 mnd1id 18793 gsumws1 18851 grp1 19065 dprdsn 20056 ring1 20307 ixpsnbasval 21215 frgpcyg 21592 mat1dimscm 22481 mat1dimmul 22482 mat1rhmelval 22486 m1detdiag 22603 pt1hmeo 23814 noextenddif 27713 noextendlt 27714 noextendgt 27715 1loopgrvd0 29522 1hevtxdg0 29523 1hevtxdg1 29524 1egrvtxdg1 29527 wlkl0 30386 actfunsnrndisj 34620 reprsuc 34630 breprexplema 34645 cvmliftlem7 35296 cvmliftlem13 35301 bj-fununsn2 37255 sticksstones9 42155 sticksstones11 42157 metakunt20 42225 frlmsnic 42550 sumsnd 45031 ovnovollem1 46671 nnsum3primesprm 47777 lincvalsng 48333 snlindsntorlem 48387 lmod1lem2 48405 lmod1lem3 48406 0aryfvalelfv 48556 1arympt1fv 48560 | 
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