Theorem refimssco 37913

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 4657	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑥𝐴𝑧 ↔ 𝑥𝐴𝑥))
2		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧𝐴𝑦 ↔ 𝑥𝐴𝑦))
3	1, 2	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → ((𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦)))
4	3	biimprd 238	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦) → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
5	4	spimev 2259	. . . . . . . 8 ⊢ ((𝑥𝐴𝑥 ∧ 𝑥𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦))
6	5	ex 450	. . . . . . 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 1910	. . . 4 ⊢ (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
11	9, 10	sylibr 224	. . 3 ⊢ ((𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)) → ∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
12	11	2alimi 1740	. 2 ⊢ (∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)) → ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
13		reflexg 37911	. 2 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)))
14		cnvssco 37912	. 2 ⊢ (◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧 ∧ 𝑧𝐴𝑦)))
15	12, 13, 14	3imtr4i 281	1 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 → ◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704   ∪ cun 3572   ⊆ wss 3574   class class class wbr 4653   I cid 5023  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   ∘ ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by: (None)