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Mirrors > Home > ILE Home > Th. List > op1sta | GIF version |
Description: Extract the first member of an ordered pair. (See op2nda 4805 to extract the second member and op1stb 4209 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 4794 | . . 3 ⊢ dom {〈𝐴, 𝐵〉} = {𝐴} |
3 | 2 | unieqi 3590 | . 2 ⊢ ∪ dom {〈𝐴, 𝐵〉} = ∪ {𝐴} |
4 | cnvsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 4 | unisn 3596 | . 2 ⊢ ∪ {𝐴} = 𝐴 |
6 | 3, 5 | eqtri 2060 | 1 ⊢ ∪ dom {〈𝐴, 𝐵〉} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 Vcvv 2557 {csn 3375 〈cop 3378 ∪ cuni 3580 dom cdm 4345 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-dm 4355 |
This theorem is referenced by: op1st 5773 fo1st 5784 f1stres 5786 xpassen 6304 xpdom2 6305 |
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