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Mirrors > Home > MPE Home > Th. List > opeldm | Structured version Visualization version GIF version |
Description: Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.) |
Ref | Expression |
---|---|
opeldm.1 | ⊢ 𝐴 ∈ V |
opeldm.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opeldm | ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeldm.2 | . . 3 ⊢ 𝐵 ∈ V | |
2 | opeq2 4874 | . . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩) | |
3 | 2 | eleq1d 2818 | . . 3 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ ∈ 𝐶 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)) |
4 | 1, 3 | spcev 3596 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶) |
5 | opeldm.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | 5 | eldm2 5901 | . 2 ⊢ (𝐴 ∈ dom 𝐶 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐶) |
7 | 4, 6 | sylibr 233 | 1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 ⟨cop 4634 dom cdm 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-dm 5686 |
This theorem is referenced by: breldm 5908 elreldm 5934 relssres 6022 iss 6035 imadmrn 6069 dfco2a 6245 relssdmrn 6267 funssres 6592 funun 6594 frxp2 8129 frxp3 8136 frrlem8 8277 frrlem10 8279 tz7.48-1 8442 iiner 8782 r0weon 10006 axdc3lem2 10445 uzrdgfni 13922 imasaddfnlem 17473 imasvscafn 17482 cicsym 17750 gsum2d 19839 cffldtocusgr 28701 dfcnv2 31896 bnj1379 33836 iss2 37208 rfovcnvf1od 42745 |
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