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Mirrors > Home > MPE Home > Th. List > dmtpop | Structured version Visualization version GIF version |
Description: The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
dmsnop.1 | ⊢ 𝐵 ∈ V |
dmprop.1 | ⊢ 𝐷 ∈ V |
dmtpop.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
dmtpop | ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = {𝐴, 𝐶, 𝐸} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4553 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) | |
2 | 1 | dmeqi 5754 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = dom ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) |
3 | dmun 5760 | . . 3 ⊢ dom ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) = (dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ dom {〈𝐸, 𝐹〉}) | |
4 | dmsnop.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
5 | dmprop.1 | . . . . 5 ⊢ 𝐷 ∈ V | |
6 | 4, 5 | dmprop 6055 | . . . 4 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} |
7 | dmtpop.1 | . . . . 5 ⊢ 𝐹 ∈ V | |
8 | 7 | dmsnop 6054 | . . . 4 ⊢ dom {〈𝐸, 𝐹〉} = {𝐸} |
9 | 6, 8 | uneq12i 4120 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ dom {〈𝐸, 𝐹〉}) = ({𝐴, 𝐶} ∪ {𝐸}) |
10 | 2, 3, 9 | 3eqtri 2847 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({𝐴, 𝐶} ∪ {𝐸}) |
11 | df-tp 4553 | . 2 ⊢ {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸}) | |
12 | 10, 11 | eqtr4i 2846 | 1 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = {𝐴, 𝐶, 𝐸} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3481 ∪ cun 3917 {csn 4548 {cpr 4550 {ctp 4552 〈cop 4554 dom cdm 5536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pr 5311 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-rab 3142 df-v 3483 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-br 5048 df-dm 5546 |
This theorem is referenced by: fntp 6396 fntpb 6953 cnfldfun 20535 |
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