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Theorem idinxpssinxp2 34574
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 idinxpres 5670 . . . 4 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
21sseq1i 3823 . . 3 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
3 idrefALT 5724 . . 3 (( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
4 brinxp2 5381 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
5 pm4.24 560 . . . . . 6 (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐴))
65anbi1i 618 . . . . 5 ((𝑥𝐴𝑥𝑅𝑥) ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
74, 6bitr4i 270 . . . 4 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥𝐴𝑥𝑅𝑥))
87ralbii 3159 . . 3 (∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
92, 3, 83bitri 289 . 2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
10 ralanid 3097 . 2 (∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
119, 10bitri 267 1 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  wcel 2157  wral 3087  cin 3766  wss 3767   class class class wbr 4841   I cid 5217   × cxp 5308  cres 5312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-res 5322
This theorem is referenced by:  idinxpssinxp3  34575  idinxpssinxp4  34576
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