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| 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 4631 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) | |
| 2 | 1 | dmeqi 5915 | . . 3 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = dom ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉}) | 
| 3 | dmun 5921 | . . 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 6237 | . . . 4 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {𝐴, 𝐶} | 
| 7 | dmtpop.1 | . . . . 5 ⊢ 𝐹 ∈ V | |
| 8 | 7 | dmsnop 6236 | . . . 4 ⊢ dom {〈𝐸, 𝐹〉} = {𝐸} | 
| 9 | 6, 8 | uneq12i 4166 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ dom {〈𝐸, 𝐹〉}) = ({𝐴, 𝐶} ∪ {𝐸}) | 
| 10 | 2, 3, 9 | 3eqtri 2769 | . 2 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = ({𝐴, 𝐶} ∪ {𝐸}) | 
| 11 | df-tp 4631 | . 2 ⊢ {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸}) | |
| 12 | 10, 11 | eqtr4i 2768 | 1 ⊢ dom {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉} = {𝐴, 𝐶, 𝐸} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 {cpr 4628 {ctp 4630 〈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-tp 4631 df-op 4633 df-br 5144 df-dm 5695 | 
| This theorem is referenced by: fntp 6627 fntpb 7229 cnfldfunALT 21379 cnfldfunALTOLD 21392 cnfldfunALTOLDOLD 21393 | 
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