![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dmpropg | Structured version Visualization version 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 6224 | . . 3 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
2 | dmsnopg 6224 | . . 3 ⊢ (𝐷 ∈ 𝑊 → dom {〈𝐶, 𝐷〉} = {𝐶}) | |
3 | uneq12 4158 | . . 3 ⊢ ((dom {〈𝐴, 𝐵〉} = {𝐴} ∧ dom {〈𝐶, 𝐷〉} = {𝐶}) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) | |
4 | 1, 2, 3 | syl2an 594 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) |
5 | df-pr 4636 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
6 | 5 | dmeqi 5911 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) |
7 | dmun 5917 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) | |
8 | 6, 7 | eqtri 2754 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) |
9 | df-pr 4636 | . 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶}) | |
10 | 4, 8, 9 | 3eqtr4g 2791 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∪ cun 3945 {csn 4633 {cpr 4635 〈cop 4639 dom cdm 5682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-dm 5692 |
This theorem is referenced by: dmprop 6228 funtpg 6614 fnprg 6618 hashdmpropge2 14502 s2dmALT 14917 s4dom 14928 estrreslem2 18162 structiedg0val 28958 |
Copyright terms: Public domain | W3C validator |