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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 6553 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {⟨𝐴, 𝐵⟩}) | |
2 | opex 5422 | . . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V | |
3 | 2 | snid 4623 | . 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} |
4 | funopfv 6895 | . 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 397 = wceq 1542 ∈ wcel 2107 {csn 4587 ⟨cop 4593 Fun wfun 6491 ‘cfv 6497 |
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 5257 ax-nul 5264 ax-pr 5385 |
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 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 |
This theorem is referenced by: fvsn 7128 fvsnun1 7129 fsnunfv 7134 fvpr1g 7137 fvpr2gOLD 7139 fsnex 7230 suppsnop 8110 mapsnend 8983 enfixsn 9028 axdc3lem4 10394 fseq1p1m1 13521 1fv 13566 s1fv 14504 sumsnf 15633 prodsn 15850 prodsnf 15852 seq1st 16452 vdwlem8 16865 setsid 17085 mgm1 18518 sgrp1 18560 mnd1 18602 mnd1id 18603 gsumws1 18653 grp1 18859 dprdsn 19820 ring1 20031 ixpsnbasval 20695 frgpcyg 20996 mat1dimscm 21840 mat1dimmul 21841 mat1rhmelval 21845 m1detdiag 21962 pt1hmeo 23173 noextenddif 27032 noextendlt 27033 noextendgt 27034 1loopgrvd0 28494 1hevtxdg0 28495 1hevtxdg1 28496 1egrvtxdg1 28499 wlkl0 29353 actfunsnrndisj 33275 reprsuc 33285 breprexplema 33300 cvmliftlem7 33942 cvmliftlem13 33947 bj-fununsn2 35771 sticksstones9 40608 sticksstones11 40610 metakunt20 40642 frlmsnic 40771 sumsnd 43319 ovnovollem1 44983 nnsum3primesprm 46068 lincvalsng 46583 snlindsntorlem 46637 lmod1lem2 46655 lmod1lem3 46656 0aryfvalelfv 46807 1arympt1fv 46811 |
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