Theorem idinxpss 38313

Description: Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.)

Assertion

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

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

Proof of Theorem idinxpss

Step	Hyp	Ref	Expression
1		inxpss 38312	. 2 ⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 I 𝑦 → 𝑥𝑅𝑦))
2		ideqg 5862	. . . . 5 ⊢ (𝑦 ∈ V → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
3	2	elv 3485	. . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4	3	imbi1i 349	. . 3 ⊢ ((𝑥 I 𝑦 → 𝑥𝑅𝑦) ↔ (𝑥 = 𝑦 → 𝑥𝑅𝑦))
5	4	2ralbii 3128	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 I 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦))
6	1, 5	bitri 275	1 ⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦))

Syntax hints:   → wi 4   ↔ wb 206  ∀wral 3061  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951   class class class wbr 5143   I cid 5577   × cxp 5683

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692

This theorem is referenced by:  ref5  38314  refrelcoss2  38465  dfrefrels3  38515  dfrefrel3  38517  symrefref3  38565