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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 6181 to extract the second member, op1stb 5429 for an alternate version, and op1st 7930 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 6169 | . . 3 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴} |
3 | 2 | unieqi 4879 | . 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = ∪ {𝐴} |
4 | cnvsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 4 | unisn 4888 | . 2 ⊢ ∪ {𝐴} = 𝐴 |
6 | 3, 5 | eqtri 2765 | 1 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3446 {csn 4587 ⟨cop 4593 ∪ cuni 4866 dom cdm 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-dm 5644 |
This theorem is referenced by: elxp4 7860 op1st 7930 fo1st 7942 f1stres 7946 xpassen 9011 xpdom2 9012 |
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