Theorem idinxpssinxp2 39350

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)) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem idinxpssinxp2

Step	Hyp	Ref	Expression
1		idinxpresid 6039	. . . 4 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
2	1	sseq1i 3936	. . 3 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
3		idrefALT 6106	. . 3 ⊢ (( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
4		brinxp2 5728	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
5		pm4.24 574	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6	5	anbi1i 636	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
7	4, 6	bitr4i 281	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥))
8	7	ralbii 3108	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥))
9	2, 3, 8	3bitri 300	. 2 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥))
10		ralanid 3110	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
11	9, 10	bitri 278	1 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∩ cin 4089   class class class wbr 5102   I cid 5544   × cxp 5648   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-res 5662

This theorem is used by:  idinxpssinxp3  39351  idinxpssinxp4  39352