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Theorem ref5 36526
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 2019 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
21imbi1i 350 . . . . 5 ((𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑥𝑅𝑦))
32ralbii 3092 . . . 4 (∀𝑦𝐵 (𝑦 = 𝑥𝑥𝑅𝑦) ↔ ∀𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
4 breq2 5085 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
54ceqsralv2TEMP 36525 . . . 4 (∀𝑦𝐵 (𝑦 = 𝑥𝑥𝑅𝑦) ↔ (𝑥𝐵𝑥𝑅𝑥))
63, 5bitr3i 277 . . 3 (∀𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ (𝑥𝐵𝑥𝑅𝑥))
76ralbii 3092 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 (𝑥𝐵𝑥𝑅𝑥))
8 idinxpss 36523 . 2 (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦𝑥𝑅𝑦))
9 ralin 36456 . 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 1539  wcel 2104  wral 3061  cin 3891  wss 3892   class class class wbr 5081   I cid 5499   × cxp 5598
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3333  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-id 5500  df-xp 5606  df-rel 5607
This theorem is referenced by:  dfrefrel5  36731
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