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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 7127 | . 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 3444 {csn 4587 ⟨cop 4593 ‘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: fvpr1OLD 7141 frrlem12 8229 elixpsn 8878 ac6sfi 9234 dcomex 10388 axdc3lem4 10394 0ram 16897 mdet0fv0 21959 chpmat0d 22199 imasdsf1olem 23742 axlowdimlem8 27940 axlowdimlem11 27943 subfacp1lem2a 33831 subfacp1lem5 33835 cvmliftlem4 33939 finixpnum 36109 poimirlem3 36127 fdc 36250 grposnOLD 36387 1arymaptfo 46815 mndtchom 47196 mndtcco 47197 |
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