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Theorem idinxpssinxp2 38319
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 6066 . . . 4 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
21sseq1i 4012 . . 3 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
3 idrefALT 6131 . . 3 (( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
4 brinxp2 5763 . . . . 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 3093 . . 3 (∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
92, 3, 83bitri 297 . 2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
10 ralanid 3095 . 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 2108  wral 3061  cin 3950  wss 3951   class class class wbr 5143   I cid 5577   × cxp 5683  cres 5687
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-res 5697
This theorem is referenced by:  idinxpssinxp3  38320  idinxpssinxp4  38321
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