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Mirrors > Home > MPE Home > Th. List > op1sta | Structured version Visualization version GIF version |
Description: Extract the first member of an ordered pair. (See op2nda 6250 to extract the second member, op1stb 5482 for an alternate version, and op1st 8021 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.) |
Ref | Expression |
---|---|
cnvsn.1 | ⊢ 𝐴 ∈ V |
cnvsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
op1sta | ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvsn.2 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | 1 | dmsnop 6238 | . . 3 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
3 | 2 | unieqi 4924 | . 2 ⊢ ∪ dom {〈𝐴, 𝐵〉} = ∪ {𝐴} |
4 | cnvsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 4 | unisn 4931 | . 2 ⊢ ∪ {𝐴} = 𝐴 |
6 | 3, 5 | eqtri 2763 | 1 ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 ∪ cuni 4912 dom cdm 5689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-dm 5699 |
This theorem is referenced by: elxp4 7945 op1st 8021 fo1st 8033 f1stres 8037 xpassen 9105 xpdom2 9106 |
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