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Theorem relssdmrnOLD 6267
Description: Obsolete version of relssdmrn 6266 as of 23-Dec-2024. (Contributed by NM, 3-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
relssdmrnOLD (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Proof of Theorem relssdmrnOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (Rel 𝐴 → Rel 𝐴)
2 19.8a 2167 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
3 19.8a 2167 . . . 4 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
4 opelxp 5708 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴))
5 vex 3473 . . . . . . 7 𝑥 ∈ V
65eldm2 5898 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
7 vex 3473 . . . . . . 7 𝑦 ∈ V
87elrn2 5889 . . . . . 6 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
96, 8anbi12i 626 . . . . 5 ((𝑥 ∈ dom 𝐴𝑦 ∈ ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
104, 9bitri 275 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴))
112, 3, 10sylanbrc 582 . . 3 (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴))
1211a1i 11 . 2 (Rel 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴)))
131, 12relssdv 5784 1 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1774  wcel 2099  wss 3944  cop 4630   × cxp 5670  dom cdm 5672  ran crn 5673  Rel wrel 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683
This theorem is referenced by: (None)
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