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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 7178 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4629 ⟨cop 4635 ‘cfv 6544 |
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 5300 ax-nul 5307 ax-pr 5428 |
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 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 |
This theorem is referenced by: fvpr1OLD 7192 frrlem12 8282 elixpsn 8931 ac6sfi 9287 dcomex 10442 axdc3lem4 10448 0ram 16953 mdet0fv0 22096 chpmat0d 22336 imasdsf1olem 23879 axlowdimlem8 28207 axlowdimlem11 28210 subfacp1lem2a 34171 subfacp1lem5 34175 cvmliftlem4 34279 finixpnum 36473 poimirlem3 36491 fdc 36613 grposnOLD 36750 1arymaptfo 47329 mndtchom 47710 mndtcco 47711 |
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