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Theorem idinxpssinxp4 37715
Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 37713). (Contributed by Peter Mazsa, 8-Mar-2019.)
Assertion
Ref Expression
idinxpssinxp4 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦

Proof of Theorem idinxpssinxp4
StepHypRef Expression
1 idinxpssinxp 37712 . 2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦))
2 idinxpssinxp2 37713 . 2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
31, 2bitr3i 277 1 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wral 3056  cin 3943  wss 3944   class class class wbr 5142   I cid 5569   × cxp 5670
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-11 2147  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
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 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-res 5684
This theorem is referenced by:  refrelcoss3  37859
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