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Theorem ref5 37120
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 2022 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
21imbi1i 350 . . . . 5 ((𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑥𝑅𝑦))
32ralbii 3094 . . . 4 (∀𝑦𝐵 (𝑦 = 𝑥𝑥𝑅𝑦) ↔ ∀𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
4 breq2 5151 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
54ceqsralbv 3644 . . . 4 (∀𝑦𝐵 (𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥𝐵𝑥𝑅𝑥))
63, 5bitr3i 277 . . 3 (∀𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ (𝑥𝐵𝑥𝑅𝑥))
76ralbii 3094 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝑅𝑥))
8 idinxpss 37119 . 2 (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
9 ralin 37053 . 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 1542  wcel 2107  wral 3062  cin 3946  wss 3947   class class class wbr 5147   I cid 5572   × cxp 5673
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682
This theorem is referenced by:  dfrefrel5  37325
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