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Mirrors > Home > NFE Home > Th. List > op1sta | GIF version |
Description: Extract the first member of an ordered pair. (Contributed by Raph Levien, 4-Dec-2003.) |
Ref | Expression |
---|---|
op1sta.1 | ⊢ A ∈ V |
op1sta.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
op1sta | ⊢ ∪dom {〈A, B〉} = A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op1sta.2 | . . . 4 ⊢ B ∈ V | |
2 | 1 | dmsnop 5070 | . . 3 ⊢ dom {〈A, B〉} = {A} |
3 | 2 | unieqi 3902 | . 2 ⊢ ∪dom {〈A, B〉} = ∪{A} |
4 | op1sta.1 | . . 3 ⊢ A ∈ V | |
5 | 4 | unisn 3908 | . 2 ⊢ ∪{A} = A |
6 | 3, 5 | eqtri 2373 | 1 ⊢ ∪dom {〈A, B〉} = A |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 Vcvv 2860 {csn 3738 ∪cuni 3892 〈cop 4562 dom cdm 4773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-ima 4728 df-cnv 4786 df-rn 4787 df-dm 4788 |
This theorem is referenced by: elxp4 5109 |
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