Theorem idinxpssinxp2 36441

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 5953	. . . 4 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
2	1	sseq1i 3954	. . 3 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
3		idrefALT 6016	. . 3 ⊢ (( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
4		brinxp2 5664	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
5		pm4.24 564	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6	5	anbi1i 624	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
7	4, 6	bitr4i 277	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥))
8	7	ralbii 3093	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥))
9	2, 3, 8	3bitri 297	. 2 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥))
10		ralanid 3096	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
11	9, 10	bitri 274	1 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2110  ∀wral 3066   ∩ cin 3891   ⊆ wss 3892   class class class wbr 5079   I cid 5488   × cxp 5587   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-id 5489  df-xp 5595  df-rel 5596  df-res 5601

This theorem is referenced by:  idinxpssinxp3  36442  idinxpssinxp4  36443