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Mirrors > Home > MPE Home > Th. List > dmsnopg | Structured version Visualization version GIF version |
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmsnopg | ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opth1 5367 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 → 𝑥 = 𝐴) |
4 | 3 | exlimiv 1931 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 → 𝑥 = 𝐴) |
5 | opeq1 4803 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉) | |
6 | opeq2 4804 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝐵〉) | |
7 | 6 | eqeq1d 2823 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉)) |
8 | 7 | spcegv 3597 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → (〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉 → ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉)) |
9 | 5, 8 | syl5 34 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥 = 𝐴 → ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉)) |
10 | 4, 9 | impbid2 228 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 𝑥 = 𝐴)) |
11 | 1 | eldm2 5770 | . . . 4 ⊢ (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉}) |
12 | opex 5356 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
13 | 12 | elsn 4582 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
14 | 13 | exbii 1848 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
15 | 11, 14 | bitri 277 | . . 3 ⊢ (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
16 | velsn 4583 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
17 | 10, 15, 16 | 3bitr4g 316 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ 𝑥 ∈ {𝐴})) |
18 | 17 | eqrdv 2819 | 1 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {csn 4567 〈cop 4573 dom cdm 5555 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-dm 5565 |
This theorem is referenced by: dmsnopss 6071 dmpropg 6072 dmsnop 6073 rnsnopg 6078 fnsng 6406 funprg 6408 funtpg 6409 fntpg 6414 funsnfsupp 8857 s1dmALT 13963 setsval 16513 setsdm 16517 estrreslem2 17388 snstriedgval 26823 1loopgrvd0 27286 1hevtxdg0 27287 1hevtxdg1 27288 1egrvtxdg1 27291 p1evtxdeqlem 27294 wlkp1 27463 eupthp1 27995 trlsegvdeglem5 28003 cosnopne 30430 bnj96 32137 bnj535 32162 ovnovollem1 42958 |
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