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Theorem idrefALT 6106
Description: Alternate proof of idref 7140 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 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Distinct variable groups:   𝑥,𝑅   𝑥,𝐴

Proof of Theorem idrefALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3963 . 2 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅))
2 elrid 6039 . . . . . 6 (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩)
32imbi1i 349 . . . . 5 ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ (∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅))
4 r19.23v 3176 . . . . 5 (∀𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅) ↔ (∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅))
5 eleq1 2815 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦𝑅 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅))
6 df-br 5142 . . . . . . . 8 (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
75, 6bitr4di 289 . . . . . . 7 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦𝑅𝑥𝑅𝑥))
87pm5.74i 271 . . . . . 6 ((𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅) ↔ (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
98ralbii 3087 . . . . 5 (∀𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅) ↔ ∀𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
103, 4, 93bitr2i 299 . . . 4 ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ ∀𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
1110albii 1813 . . 3 (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ ∀𝑦𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
12 ralcom4 3277 . . 3 (∀𝑥𝐴𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑦𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
13 opex 5457 . . . . 5 𝑥, 𝑥⟩ ∈ V
14 biidd 262 . . . . 5 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑥𝑅𝑥𝑥𝑅𝑥))
1513, 14ceqsalv 3506 . . . 4 (∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
1615ralbii 3087 . . 3 (∀𝑥𝐴𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
1711, 12, 163bitr2i 299 . 2 (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
181, 17bitri 275 1 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  wcel 2098  wral 3055  wrex 3064  wss 3943  cop 4629   class class class wbr 5141   I cid 5566  cres 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-res 5681
This theorem is referenced by:  idinxpssinxp2  37700  idinxpssinxp3  37701  symrefref3  37947  refsymrels3  37949  elrefsymrels3  37953  dfeqvrels3  37972  refrelsredund3  38017  refrelredund3  38020
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