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Mirrors > Home > MPE Home > Th. List > Mathboxes > refimssco | Structured version Visualization version GIF version |
Description: Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.) |
Ref | Expression |
---|---|
refimssco | ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 → ◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5107 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑥𝐴𝑧 ↔ 𝑥𝐴𝑥)) | |
2 | breq1 5106 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
3 | 1, 2 | anbi12d 631 | . . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦))) |
4 | 3 | biimprd 247 | . . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦) → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))) |
5 | 4 | spimevw 1998 | . . . . . . . 8 ⊢ ((𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)) |
6 | 5 | ex 413 | . . . . . . 7 ⊢ (𝑥𝐴𝑥 → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))) |
7 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦) → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))) |
8 | 7 | com12 32 | . . . . 5 ⊢ (𝑥𝐴𝑦 → ((𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))) |
9 | 8 | a2i 14 | . . . 4 ⊢ ((𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)) → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))) |
10 | 19.37v 1995 | . . . 4 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))) | |
11 | 9, 10 | sylibr 233 | . . 3 ⊢ ((𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)) → ∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))) |
12 | 11 | 2alimi 1814 | . 2 ⊢ (∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)) → ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))) |
13 | reflexg 41819 | . 2 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦))) | |
14 | cnvssco 41820 | . 2 ⊢ (◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))) | |
15 | 12, 13, 14 | 3imtr4i 291 | 1 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 → ◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1539 ∃wex 1781 ∪ cun 3906 ⊆ wss 3908 class class class wbr 5103 I cid 5528 ◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 ∘ ccom 5635 |
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-11 2154 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
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 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 |
This theorem is referenced by: (None) |
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