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Mirrors > Home > ILE Home > Th. List > dmpropg | GIF version |
Description: The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmpropg | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmsnopg 5080 | . . 3 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
2 | dmsnopg 5080 | . . 3 ⊢ (𝐷 ∈ 𝑊 → dom {〈𝐶, 𝐷〉} = {𝐶}) | |
3 | uneq12 3276 | . . 3 ⊢ ((dom {〈𝐴, 𝐵〉} = {𝐴} ∧ dom {〈𝐶, 𝐷〉} = {𝐶}) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) | |
4 | 1, 2, 3 | syl2an 287 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) |
5 | df-pr 3588 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
6 | 5 | dmeqi 4810 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) |
7 | dmun 4816 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) | |
8 | 6, 7 | eqtri 2191 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) |
9 | df-pr 3588 | . 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶}) | |
10 | 4, 8, 9 | 3eqtr4g 2228 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 ∪ cun 3119 {csn 3581 {cpr 3582 〈cop 3584 dom cdm 4609 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-dm 4619 |
This theorem is referenced by: dmprop 5083 funtpg 5247 fnprg 5251 |
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