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Theorem idinxpssinxp2 35613
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 5888 . . . 4 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
21sseq1i 3971 . . 3 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
3 idrefALT 5946 . . 3 (( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
4 brinxp2 5602 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
5 pm4.24 567 . . . . . 6 (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐴))
65anbi1i 626 . . . . 5 ((𝑥𝐴𝑥𝑅𝑥) ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
74, 6bitr4i 281 . . . 4 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥𝐴𝑥𝑅𝑥))
87ralbii 3153 . . 3 (∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
92, 3, 83bitri 300 . 2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
10 ralanid 3156 . 2 (∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
119, 10bitri 278 1 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2115  wral 3126  cin 3909  wss 3910   class class class wbr 5039   I cid 5432   × cxp 5526  cres 5530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-res 5540
This theorem is referenced by:  idinxpssinxp3  35614  idinxpssinxp4  35615
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