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Theorem idrefALT 6018
Description: Alternate proof of idref 7018 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 3907 . 2 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅))
2 elrid 5953 . . . . . 6 (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩)
32imbi1i 350 . . . . 5 ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ (∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅))
4 r19.23v 3208 . . . . 5 (∀𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅) ↔ (∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅))
5 eleq1 2826 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦𝑅 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅))
6 df-br 5075 . . . . . . . 8 (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
75, 6bitr4di 289 . . . . . . 7 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦𝑅𝑥𝑅𝑥))
87pm5.74i 270 . . . . . 6 ((𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅) ↔ (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
98ralbii 3092 . . . . 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 1822 . . 3 (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ ∀𝑦𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
12 ralcom4 3164 . . 3 (∀𝑥𝐴𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑦𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
13 opex 5379 . . . . 5 𝑥, 𝑥⟩ ∈ V
14 biidd 261 . . . . 5 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑥𝑅𝑥𝑥𝑅𝑥))
1513, 14ceqsalv 3467 . . . 4 (∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
1615ralbii 3092 . . 3 (∀𝑥𝐴𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
1711, 12, 163bitr2i 299 . 2 (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
181, 17bitri 274 1 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537   = wceq 1539  wcel 2106  wral 3064  wrex 3065  wss 3887  cop 4567   class class class wbr 5074   I cid 5488  cres 5591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-res 5601
This theorem is referenced by:  idinxpssinxp2  36453  idinxpssinxp3  36454  symrefref3  36678  refsymrels3  36680  elrefsymrels3  36684  dfeqvrels3  36702  refrelsredund3  36747  refrelredund3  36750
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