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Theorem idrefALT 6131
Description: Alternate proof of idref 7166 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 df-ss 3968 . 2 (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅))
2 elrid 6064 . . . . . 6 (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩)
32imbi1i 349 . . . . 5 ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ (∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅))
4 r19.23v 3183 . . . . 5 (∀𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅) ↔ (∃𝑥𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦𝑅))
5 eleq1 2829 . . . . . . . 8 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦𝑅 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅))
6 df-br 5144 . . . . . . . 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 3093 . . . . 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 1819 . . 3 (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦𝑅) ↔ ∀𝑦𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
12 ralcom4 3286 . . 3 (∀𝑥𝐴𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑦𝑥𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
13 opex 5469 . . . . 5 𝑥, 𝑥⟩ ∈ V
14 biidd 262 . . . . 5 (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑥𝑅𝑥𝑥𝑅𝑥))
1513, 14ceqsalv 3521 . . . 4 (∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
1615ralbii 3093 . . 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 206  wal 1538   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951  cop 4632   class class class wbr 5143   I cid 5577  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-res 5697
This theorem is referenced by:  idinxpssinxp2  38319  idinxpssinxp3  38320  symrefref3  38565  refsymrels3  38567  elrefsymrels3  38571  dfeqvrels3  38590  refrelsredund3  38635  refrelredund3  38638
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