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Theorem idinxpssinxp2 34413
 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.)
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 34412 . . . 4 ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
21sseq1i 3770 . . 3 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
3 issref 5667 . . 3 (( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
4 brinxp2ALTV 34358 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
5 pm4.24 678 . . . . . 6 (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐴))
65anbi1i 733 . . . . 5 ((𝑥𝐴𝑥𝑅𝑥) ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
74, 6bitr4i 267 . . . 4 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥𝐴𝑥𝑅𝑥))
87ralbii 3118 . . 3 (∀𝑥𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
92, 3, 83bitri 286 . 2 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥))
10 ralanid 34335 . 2 (∀𝑥𝐴 (𝑥𝐴𝑥𝑅𝑥) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
119, 10bitri 264 1 (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   ∈ wcel 2139  ∀wral 3050   ∩ cin 3714   ⊆ wss 3715   class class class wbr 4804   I cid 5173   × cxp 5264   ↾ cres 5268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-iun 4674  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-fun 6051  df-fn 6052 This theorem is referenced by:  idinxpssinxp3  34414  idinxpssinxp4  34415
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