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Theorem idrefALT 6112
Description: Alternate proof of idref 7146 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 3968 . 2 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅))
2 elrid 6045 . . . . . 6 (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩)
32imbi1i 349 . . . . 5 ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ (∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅))
4 r19.23v 3181 . . . . 5 (∀𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅) ↔ (∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅))
5 eleq1 2820 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦𝑅 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅))
6 df-br 5149 . . . . . . . 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 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 1820 . . 3 (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ ∀𝑦𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
12 ralcom4 3282 . . 3 (∀𝑥𝐴𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑦𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
13 opex 5464 . . . . 5 𝑥, 𝑥⟩ ∈ V
14 biidd 262 . . . . 5 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑥𝑅𝑥𝑥𝑅𝑥))
1513, 14ceqsalv 3511 . . . 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 275 1 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1538   = wceq 1540  wcel 2105  wral 3060  wrex 3069  wss 3948  cop 4634   class class class wbr 5148   I cid 5573  cres 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-res 5688
This theorem is referenced by:  idinxpssinxp2  37651  idinxpssinxp3  37652  symrefref3  37898  refsymrels3  37900  elrefsymrels3  37904  dfeqvrels3  37923  refrelsredund3  37968  refrelredund3  37971
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