Theorem idinxpssinxp2 38691

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 6000	. . . 4 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
2	1	sseq1i 3943	. . 3 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
3		idrefALT 6063	. . 3 ⊢ (( I ↾ 𝐴) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
4		brinxp2 5696	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
5		pm4.24 568	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
6	5	anbi1i 630	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥𝑅𝑥))
7	4, 6	bitr4i 279	. . . 4 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥))
8	7	ralbii 3085	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥))
9	2, 3, 8	3bitri 298	. 2 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥))
10		ralanid 3087	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
11	9, 10	bitri 276	1 ⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  ∀wral 3053   ∩ cin 3882   ⊆ wss 3883   class class class wbr 5072   I cid 5512   × cxp 5616   ↾ cres 5620

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-res 5630

This theorem is referenced by:  idinxpssinxp3  38692  idinxpssinxp4  38693