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Theorem ref5 37780
Description: Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 12-Dec-2023.)
Assertion
Ref Expression
ref5 (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ (𝐴𝐵)𝑥𝑅𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem ref5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 equcom 2014 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
21imbi1i 349 . . . . 5 ((𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑥𝑅𝑦))
32ralbii 3089 . . . 4 (∀𝑦𝐵 (𝑦 = 𝑥𝑥𝑅𝑦) ↔ ∀𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
4 breq2 5147 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
54ceqsralbv 3642 . . . 4 (∀𝑦𝐵 (𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥𝐵𝑥𝑅𝑥))
63, 5bitr3i 277 . . 3 (∀𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ (𝑥𝐵𝑥𝑅𝑥))
76ralbii 3089 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝑅𝑥))
8 idinxpss 37779 . 2 (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
9 ralin 37714 . 2 (∀𝑥 ∈ (𝐴𝐵)𝑥𝑅𝑥 ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝑅𝑥))
107, 8, 93bitr4i 303 1 (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ (𝐴𝐵)𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  wral 3057  cin 3944  wss 3945   class class class wbr 5143   I cid 5570   × cxp 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-id 5571  df-xp 5679  df-rel 5680
This theorem is referenced by:  dfrefrel5  37984
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