![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > op1sta | GIF version |
Description: Extract the first member of an ordered pair. (See op2nda 5031 to extract the second member and op1stb 4407 for an alternate 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 5020 | . . 3 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
3 | 2 | unieqi 3754 | . 2 ⊢ ∪ dom {〈𝐴, 𝐵〉} = ∪ {𝐴} |
4 | cnvsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 4 | unisn 3760 | . 2 ⊢ ∪ {𝐴} = 𝐴 |
6 | 3, 5 | eqtri 2161 | 1 ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 Vcvv 2689 {csn 3532 〈cop 3535 ∪ cuni 3744 dom cdm 4547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-dm 4557 |
This theorem is referenced by: op1st 6052 fo1st 6063 f1stres 6065 xpassen 6732 xpdom2 6733 |
Copyright terms: Public domain | W3C validator |