Theorem relssdmrnOLD 6268

Description: Obsolete version of relssdmrn 6267 as of 23-Dec-2024. (Contributed by NM, 3-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
relssdmrnOLD	⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Proof of Theorem relssdmrnOLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (Rel 𝐴 → Rel 𝐴)
2		19.8a 2174	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
3		19.8a 2174	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
4		opelxp 5712	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴))
5		vex 3478	. . . . . . 7 ⊢ 𝑥 ∈ V
6	5	eldm2 5901	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
7		vex 3478	. . . . . . 7 ⊢ 𝑦 ∈ V
8	7	elrn2 5892	. . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
9	6, 8	anbi12i 627	. . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴))
10	4, 9	bitri 274	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴))
11	2, 3, 10	sylanbrc 583	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴))
12	11	a1i 11	. 2 ⊢ (Rel 𝐴 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴)))
13	1, 12	relssdv 5788	1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1781   ∈ wcel 2106   ⊆ wss 3948  ⟨cop 4634   × cxp 5674  dom cdm 5676  ran crn 5677  Rel wrel 5681

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-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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-ral 3062  df-rex 3071  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-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687

This theorem is referenced by: (None)