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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 3406 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) | |
2 | 1 | dmeqi 4554 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = dom ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) |
3 | dmun 4560 | . . 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 4815 | . . . 4 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} |
7 | dmtpop.1 | . . . . 5 ⊢ 𝐹 ∈ V | |
8 | 7 | dmsnop 4814 | . . . 4 ⊢ dom {〈𝐸, 𝐹〉} = {𝐸} |
9 | 6, 8 | uneq12i 3124 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ dom {〈𝐸, 𝐹〉}) = ({𝐴, 𝐶} ∪ {𝐸}) |
10 | 2, 3, 9 | 3eqtri 2105 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({𝐴, 𝐶} ∪ {𝐸}) |
11 | df-tp 3406 | . 2 ⊢ {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸}) | |
12 | 10, 11 | eqtr4i 2104 | 1 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = {𝐴, 𝐶, 𝐸} |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 Vcvv 2601 ∪ cun 2971 {csn 3398 {cpr 3399 {ctp 3400 〈cop 3401 dom cdm 4363 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-tp 3406 df-op 3407 df-br 3786 df-dm 4373 |
This theorem is referenced by: fntp 4976 |
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