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Theorem ref5 38294
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 3090 . . . 4 (∀𝑦𝐵 (𝑦 = 𝑥𝑥𝑅𝑦) ↔ ∀𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
4 breq2 5151 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
54ceqsralbv 3656 . . . 4 (∀𝑦𝐵 (𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥𝐵𝑥𝑅𝑥))
63, 5bitr3i 277 . . 3 (∀𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ (𝑥𝐵𝑥𝑅𝑥))
76ralbii 3090 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝑅𝑥))
8 idinxpss 38293 . 2 (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
9 ralin 38229 . 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 206   = wceq 1536  wcel 2105  wral 3058  cin 3961  wss 3962   class class class wbr 5147   I cid 5581   × cxp 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695
This theorem is referenced by:  dfrefrel5  38498
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