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Theorem idinxpssinxp2 38299
Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 4-Mar-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
idinxpssinxp2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem idinxpssinxp2
StepHypRef Expression
1 idinxpresid 6067 . . . 4 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
21sseq1i 4023 . . 3 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
3 idrefALT 6133 . . 3 (( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
4 brinxp2 5765 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
5 pm4.24 563 . . . . . 6 (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐴))
65anbi1i 624 . . . . 5 ((𝑥𝐴𝑥𝑅𝑥) ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
74, 6bitr4i 278 . . . 4 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥𝐴𝑥𝑅𝑥))
87ralbii 3090 . . 3 (∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
92, 3, 83bitri 297 . 2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
10 ralanid 3092 . 2 (∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
119, 10bitri 275 1 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2105  wral 3058  cin 3961  wss 3962   class class class wbr 5147   I cid 5581   × cxp 5686  cres 5690
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-11 2154  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  df-res 5700
This theorem is referenced by:  idinxpssinxp3  38300  idinxpssinxp4  38301
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