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 5524 | . . 3 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
2 | dmsnopg 5524 | . . 3 ⊢ (𝐷 ∈ 𝑊 → dom {〈𝐶, 𝐷〉} = {𝐶}) | |
3 | uneq12 3724 | . . 3 ⊢ ((dom {〈𝐴, 𝐵〉} = {𝐴} ∧ dom {〈𝐶, 𝐷〉} = {𝐶}) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) | |
4 | 1, 2, 3 | syl2an 493 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) |
5 | df-pr 4128 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
6 | 5 | dmeqi 5247 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) |
7 | dmun 5253 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) | |
8 | 6, 7 | eqtri 2632 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) |
9 | df-pr 4128 | . 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶}) | |
10 | 4, 8, 9 | 3eqtr4g 2669 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 {csn 4125 {cpr 4127 〈cop 4131 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-dm 5048 |
This theorem is referenced by: dmprop 5528 funtpg 5856 funtpgOLD 5857 fnprg 5861 s2dmALT 13503 s4dom 13514 estrreslem2 16601 structvtxvallem 25697 structiedg0val 25699 structgrssvtxlem 25700 |
Copyright terms: Public domain | W3C validator |