Theorem refimssco 41104

Description: Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.)

Assertion

Ref	Expression
refimssco	⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 → ◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴))

Proof of Theorem refimssco

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

Step	Hyp	Ref	Expression
1		breq2 5074	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑥𝐴𝑧 ↔ 𝑥𝐴𝑥))
2		breq1 5073	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
3	1, 2	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦)))
4	3	biimprd 247	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦) → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
5	4	spimevw 1999	. . . . . . . 8 ⊢ ((𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))
6	5	ex 412	. . . . . . 7 ⊢ (𝑥𝐴𝑥 → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
7	6	adantr 480	. . . . . 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 1996	. . . 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 1816	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)) → ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
13		reflexg 41102	. 2 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)))
14		cnvssco 41103	. 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))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1783   ∪ cun 3881   ⊆ wss 3883   class class class wbr 5070   I cid 5479  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   ∘ ccom 5584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592

This theorem is referenced by: (None)