Theorem idrefALT 6007

Description: Alternate proof of idref 7000 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		dfss2 3903	. 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
2		elrid 5942	. . . . . 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 3207	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅))
5		eleq1 2826	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦 ∈ 𝑅 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅))
6		df-br 5071	. . . . . . . 8 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
7	5, 6	bitr4di 288	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦 ∈ 𝑅 ↔ 𝑥𝑅𝑥))
8	7	pm5.74i 270	. . . . . 6 ⊢ ((𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
9	8	ralbii 3090	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
10	3, 4, 9	3bitr2i 298	. . . 4 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
11	10	albii 1823	. . 3 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
12		ralcom4 3161	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
13		opex 5373	. . . . 5 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
14		biidd 261	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑥))
15	13, 14	ceqsalv 3457	. . . 4 ⊢ (∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
16	15	ralbii 3090	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
17	11, 12, 16	3bitr2i 298	. 2 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
18	1, 17	bitri 274	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ⟨cop 4564   class class class wbr 5070   I cid 5479   ↾ cres 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-res 5592

This theorem is referenced by:  idinxpssinxp2  36380  idinxpssinxp3  36381  symrefref3  36605  refsymrels3  36607  elrefsymrels3  36611  dfeqvrels3  36629  refrelsredund3  36674  refrelredund3  36677