Theorem inxpss3 38307

Description: Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss 38304). (Contributed by Peter Mazsa, 8-Mar-2019.)

Assertion

Ref	Expression
inxpss3	⊢ (∀𝑥∀𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 → 𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥𝑆𝑦))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)

Proof of Theorem inxpss3

Step	Hyp	Ref	Expression
1		brinxp2 5701	. . . . 5 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
2		brinxp2 5701	. . . . 5 ⊢ (𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑆𝑦))
3	1, 2	imbi12i 350	. . . 4 ⊢ ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 → 𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑆𝑦)))
4		imdistan 567	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 → 𝑥𝑆𝑦)) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑆𝑦)))
5	3, 4	bitr4i 278	. . 3 ⊢ ((𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 → 𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 → 𝑥𝑆𝑦)))
6	5	2albii 1820	. 2 ⊢ (∀𝑥∀𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 → 𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 → 𝑥𝑆𝑦)))
7		r2al 3165	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥𝑆𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 → 𝑥𝑆𝑦)))
8	6, 7	bitr4i 278	1 ⊢ (∀𝑥∀𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 → 𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥𝑆𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  ∀wral 3044   ∩ cin 3904   class class class wbr 5095   × cxp 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629

This theorem is referenced by:  inxpss2  38308