Theorem inxpssidinxp 38759

Description: Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 38758. (Contributed by Peter Mazsa, 4-Jul-2019.)

Assertion

Ref	Expression
inxpssidinxp	⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem inxpssidinxp

Step	Hyp	Ref	Expression
1		inxpss2 38758	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 I 𝑦))
2		ideqg 5812	. . . . 5 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
3	2	elv 3449	. . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4	3	imbi2i 338	. . 3 ⊢ ((𝑥𝑅𝑦 → 𝑥 I 𝑦) ↔ (𝑥𝑅𝑦 → 𝑥 = 𝑦))
5	4	2ralbii 3127	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 I 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦))
6	1, 5	bitri 277	1 ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 208  ∀wral 3066  Vcvv 3444   ∩ cin 3894   ⊆ wss 3895   class class class wbr 5090   I cid 5530   × cxp 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-br 5091  df-opab 5153  df-id 5531  df-xp 5642  df-rel 5643

This theorem is referenced by:  dfcnvrefrels3  39046  dfcnvrefrel3  39048