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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.) |
Ref | Expression |
---|---|
fvsn.1 | ⊢ 𝐴 ∈ V |
fvsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fvsn | ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | fvsn.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | funsn 6082 | . 2 ⊢ Fun {〈𝐴, 𝐵〉} |
4 | opex 5060 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
5 | 4 | snid 4347 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} |
6 | funopfv 6376 | . 2 ⊢ (Fun {〈𝐴, 𝐵〉} → (〈𝐴, 𝐵〉 ∈ {〈𝐴, 𝐵〉} → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵)) | |
7 | 3, 5, 6 | mp2 9 | 1 ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∈ wcel 2145 Vcvv 3351 {csn 4316 〈cop 4322 Fun wfun 6025 ‘cfv 6031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-iota 5994 df-fun 6033 df-fv 6039 |
This theorem is referenced by: fvsng 6591 fvsnun1 6592 fvpr1 6600 elixpsn 8101 ac6sfi 8360 dcomex 9471 axdc3lem4 9477 0ram 15931 mdet0fv0 20618 chpmat0d 20859 imasdsf1olem 22398 axlowdimlem8 26050 axlowdimlem11 26053 subfacp1lem2a 31500 subfacp1lem5 31504 cvmliftlem4 31608 finixpnum 33727 poimirlem3 33745 fdc 33873 grposnOLD 34013 |
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