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Mirrors > Home > MPE Home > Th. List > relssdmrnOLD | Structured version Visualization version GIF version |
Description: Obsolete version of relssdmrn 6221 as of 23-Dec-2024. (Contributed by NM, 3-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
relssdmrnOLD | ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (Rel 𝐴 → Rel 𝐴) | |
2 | 19.8a 2175 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
3 | 19.8a 2175 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
4 | opelxp 5670 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴)) | |
5 | vex 3448 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | 5 | eldm2 5858 | . . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴) |
7 | vex 3448 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
8 | 7 | elrn2 5849 | . . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴) |
9 | 6, 8 | anbi12i 628 | . . . . 5 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ ran 𝐴) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)) |
10 | 4, 9 | bitri 275 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (dom 𝐴 × ran 𝐴) ↔ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)) |
11 | 2, 3, 10 | sylanbrc 584 | . . 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 5745 | 1 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 ⊆ wss 3911 ⟨cop 4593 × cxp 5632 dom cdm 5634 ran crn 5635 Rel wrel 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 |
This theorem is referenced by: (None) |
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