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| 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 6233 | . . 3 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
| 2 | dmsnopg 6233 | . . 3 ⊢ (𝐷 ∈ 𝑊 → dom {〈𝐶, 𝐷〉} = {𝐶}) | |
| 3 | uneq12 4163 | . . 3 ⊢ ((dom {〈𝐴, 𝐵〉} = {𝐴} ∧ dom {〈𝐶, 𝐷〉} = {𝐶}) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) | |
| 4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) |
| 5 | df-pr 4629 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
| 6 | 5 | dmeqi 5915 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) |
| 7 | dmun 5921 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) | |
| 8 | 6, 7 | eqtri 2765 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) |
| 9 | df-pr 4629 | . 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶}) | |
| 10 | 4, 8, 9 | 3eqtr4g 2802 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 {cpr 4628 〈cop 4632 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 |
| This theorem is referenced by: dmprop 6237 funtpg 6621 fnprg 6625 hashdmpropge2 14522 s2dmALT 14947 s4dom 14958 estrreslem2 18183 structiedg0val 29039 |
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