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| Description: Extract the first member of an ordered pair. (See op2nda 6248 to extract the second member, op1stb 5476 for an alternate version, and op1st 8022 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 6236 | . . 3 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} | 
| 3 | 2 | unieqi 4919 | . 2 ⊢ ∪ dom {〈𝐴, 𝐵〉} = ∪ {𝐴} | 
| 4 | cnvsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 5 | 4 | unisn 4926 | . 2 ⊢ ∪ {𝐴} = 𝐴 | 
| 6 | 3, 5 | eqtri 2765 | 1 ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ∪ cuni 4907 dom cdm 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 | 
| This theorem is referenced by: elxp4 7944 op1st 8022 fo1st 8034 f1stres 8038 xpassen 9106 xpdom2 9107 | 
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