Theorem idrefALT 6115

Description: Alternate proof of idref 7144 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 3923	. 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
2		elrid 6050	. . . . . 6 ⊢ (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩)
3	2	imbi1i 352	. . . . 5 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅))
4		r19.23v 3192	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅))
5		eleq1 2851	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦 ∈ 𝑅 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅))
6		df-br 5111	. . . . . . . 8 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
7	5, 6	bitr4di 292	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦 ∈ 𝑅 ↔ 𝑥𝑅𝑥))
8	7	pm5.74i 274	. . . . . 6 ⊢ ((𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
9	8	ralbii 3111	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
10	3, 4, 9	3bitr2i 302	. . . 4 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
11	10	albii 1849	. . 3 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
12		ralcom4 3291	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
13		opex 5447	. . . . 5 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
14		biidd 265	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑥))
15	13, 14	ceqsalv 3494	. . . 4 ⊢ (∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
16	15	ralbii 3111	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
17	11, 12, 16	3bitr2i 302	. 2 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
18	1, 17	bitri 278	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3906  ⟨cop 4596   class class class wbr 5110   I cid 5557   ↾ cres 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-res 5675

This theorem is referenced by:  idinxpssinxp2  38951  idinxpssinxp3  38952  symrefref3  39275  refsymrels3  39277  elrefsymrels3  39281  dfeqvrels3  39300  refrelsredund3  39345  refrelredund3  39348