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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 6063 | . . 3 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
2 | dmsnopg 6063 | . . 3 ⊢ (𝐷 ∈ 𝑊 → dom {〈𝐶, 𝐷〉} = {𝐶}) | |
3 | uneq12 4131 | . . 3 ⊢ ((dom {〈𝐴, 𝐵〉} = {𝐴} ∧ dom {〈𝐶, 𝐷〉} = {𝐶}) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) | |
4 | 1, 2, 3 | syl2an 595 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) = ({𝐴} ∪ {𝐶})) |
5 | df-pr 4560 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
6 | 5 | dmeqi 5766 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) |
7 | dmun 5772 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) | |
8 | 6, 7 | eqtri 2841 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = (dom {〈𝐴, 𝐵〉} ∪ dom {〈𝐶, 𝐷〉}) |
9 | df-pr 4560 | . 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶}) | |
10 | 4, 8, 9 | 3eqtr4g 2878 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 {csn 4557 {cpr 4559 〈cop 4563 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-dm 5558 |
This theorem is referenced by: dmprop 6067 funtpg 6402 fnprg 6406 hashdmpropge2 13829 s2dmALT 14258 s4dom 14269 estrreslem2 17376 structiedg0val 26734 |
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