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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 3578 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) | |
2 | 1 | dmeqi 4799 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = dom ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) |
3 | dmun 4805 | . . 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 5072 | . . . 4 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} |
7 | dmtpop.1 | . . . . 5 ⊢ 𝐹 ∈ V | |
8 | 7 | dmsnop 5071 | . . . 4 ⊢ dom {〈𝐸, 𝐹〉} = {𝐸} |
9 | 6, 8 | uneq12i 3269 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ dom {〈𝐸, 𝐹〉}) = ({𝐴, 𝐶} ∪ {𝐸}) |
10 | 2, 3, 9 | 3eqtri 2189 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({𝐴, 𝐶} ∪ {𝐸}) |
11 | df-tp 3578 | . 2 ⊢ {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸}) | |
12 | 10, 11 | eqtr4i 2188 | 1 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = {𝐴, 𝐶, 𝐸} |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∪ cun 3109 {csn 3570 {cpr 3571 {ctp 3572 〈cop 3573 dom cdm 4598 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-tp 3578 df-op 3579 df-br 3977 df-dm 4608 |
This theorem is referenced by: fntp 5239 |
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