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Mirrors > Home > ILE Home > Th. List > dmtpop | GIF version |
Description: The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
dmsnop.1 | ⊢ 𝐵 ∈ V |
dmprop.1 | ⊢ 𝐷 ∈ V |
dmtpop.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
dmtpop | ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = {𝐴, 𝐶, 𝐸} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3535 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) | |
2 | 1 | dmeqi 4740 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = dom ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) |
3 | dmun 4746 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) = (dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ dom {〈𝐸, 𝐹〉}) | |
4 | dmsnop.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
5 | dmprop.1 | . . . . 5 ⊢ 𝐷 ∈ V | |
6 | 4, 5 | dmprop 5013 | . . . 4 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} |
7 | dmtpop.1 | . . . . 5 ⊢ 𝐹 ∈ V | |
8 | 7 | dmsnop 5012 | . . . 4 ⊢ dom {〈𝐸, 𝐹〉} = {𝐸} |
9 | 6, 8 | uneq12i 3228 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ dom {〈𝐸, 𝐹〉}) = ({𝐴, 𝐶} ∪ {𝐸}) |
10 | 2, 3, 9 | 3eqtri 2164 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({𝐴, 𝐶} ∪ {𝐸}) |
11 | df-tp 3535 | . 2 ⊢ {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸}) | |
12 | 10, 11 | eqtr4i 2163 | 1 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = {𝐴, 𝐶, 𝐸} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 {csn 3527 {cpr 3528 {ctp 3529 〈cop 3530 dom cdm 4539 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-tp 3535 df-op 3536 df-br 3930 df-dm 4549 |
This theorem is referenced by: fntp 5180 |
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