Theorem refimssco 43724

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 5097	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑥𝐴𝑧 ↔ 𝑥𝐴𝑥))
2		breq1 5096	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
3	1, 2	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦)))
4	3	biimprd 248	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦) → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
5	4	spimevw 1986	. . . . . . . 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 1998	. . . 4 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
11	9, 10	sylibr 234	. . 3 ⊢ ((𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)) → ∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
12	11	2alimi 1813	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)) → ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
13		reflexg 43722	. 2 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)))
14		cnvssco 43723	. 2 ⊢ (◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
15	12, 13, 14	3imtr4i 292	1 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 → ◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539  ∃wex 1780   ∪ cun 3896   ⊆ wss 3898   class class class wbr 5093   I cid 5513  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   ∘ ccom 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631

This theorem is referenced by: (None)