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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 6078 to extract the second member, op1stb 5354 for an alternate version, and op1st 7686 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 6066 | . . 3 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
3 | 2 | unieqi 4839 | . 2 ⊢ ∪ dom {〈𝐴, 𝐵〉} = ∪ {𝐴} |
4 | cnvsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 4 | unisn 4846 | . 2 ⊢ ∪ {𝐴} = 𝐴 |
6 | 3, 5 | eqtri 2841 | 1 ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 ∪ cuni 4830 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-dm 5558 |
This theorem is referenced by: elxp4 7616 op1st 7686 fo1st 7698 f1stres 7702 xpassen 8599 xpdom2 8600 |
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