Theorem idref 7149

Description: Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.)

Assertion

Ref	Expression
idref	⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem idref

Step	Hyp	Ref	Expression
1		eqid 2760	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
2	1	fmpt 7110	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅)
3		opex 5431	. . . . 5 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
4	3, 1	fnmpti 6682	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
5		df-f 6542	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
6	4, 5	mpbiran 722	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
7	2, 6	bitri 278	. 2 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
8		df-br 5103	. . 3 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
9	8	ralbii 3108	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅)
10		mptresid 6042	. . . 4 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)
11		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
12	11	fnasrn 7148	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
13	10, 12	eqtri 2783	. . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
14	13	sseq1i 3936	. 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
15	7, 9, 14	3bitr4ri 307	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  ⟨cop 4834   class class class wbr 5102   ↦ cmpt 5185   I cid 5544  ran crn 5651   ↾ cres 5652   Fn wfn 6533  ⟶wf 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545

This theorem is used by:  retos  21929  filnetlem2  37259