Theorem idref 7095

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 2740	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
2	1	fmpt 7058	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅)
3		opex 5410	. . . . 5 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
4	3, 1	fnmpti 6635	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴
5		df-f 6496	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅 ↔ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅))
6	4, 5	mpbiran 715	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩):𝐴⟶𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
7	2, 6	bitri 276	. 2 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
8		df-br 5080	. . 3 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
9	8	ralbii 3086	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑥, 𝑥⟩ ∈ 𝑅)
10		mptresid 6010	. . . 4 ⊢ ( I ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝑥)
11		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
12	11	fnasrn 7094	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑥) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
13	10, 12	eqtri 2763	. . 3 ⊢ ( I ↾ 𝐴) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩)
14	13	sseq1i 3950	. 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝑥⟩) ⊆ 𝑅)
15	7, 9, 14	3bitr4ri 305	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   ↔ wb 207   ∈ wcel 2119  ∀wral 3054   ⊆ wss 3890  ⟨cop 4568   class class class wbr 5079   ↦ cmpt 5160   I cid 5519  ran crn 5626   ↾ cres 5627   Fn wfn 6487  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499

This theorem is referenced by:  retos  21600  filnetlem2  36614