Theorem idrefALT 6143

Description: Alternate proof of idref 7180 not relying on definitions related to functions. 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.) (Proof shortened by BJ, 28-Aug-2022.) The "proof modification is discouraged" tag is here only because this is an *ALT result. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝑅   𝑥,𝐴

Proof of Theorem idrefALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ss 3993	. 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
2		elrid 6075	. . . . . 6 ⊢ (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩)
3	2	imbi1i 349	. . . . 5 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅))
4		r19.23v 3189	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅))
5		eleq1 2832	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦 ∈ 𝑅 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅))
6		df-br 5167	. . . . . . . 8 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
7	5, 6	bitr4di 289	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦 ∈ 𝑅 ↔ 𝑥𝑅𝑥))
8	7	pm5.74i 271	. . . . . 6 ⊢ ((𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
9	8	ralbii 3099	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
10	3, 4, 9	3bitr2i 299	. . . 4 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
11	10	albii 1817	. . 3 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
12		ralcom4 3292	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
13		opex 5484	. . . . 5 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
14		biidd 262	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑥))
15	13, 14	ceqsalv 3529	. . . 4 ⊢ (∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
16	15	ralbii 3099	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
17	11, 12, 16	3bitr2i 299	. 2 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
18	1, 17	bitri 275	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ⟨cop 4654   class class class wbr 5166   I cid 5592   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-res 5712

This theorem is referenced by:  idinxpssinxp2  38274  idinxpssinxp3  38275  symrefref3  38520  refsymrels3  38522  elrefsymrels3  38526  dfeqvrels3  38545  refrelsredund3  38590  refrelredund3  38593