Theorem idssxp 28811

Description: A diagonal set as a subset of a Cartesian product. (Contributed by Thierry Arnoux, 29-Dec-2019.)

Assertion

Ref	Expression
idssxp	⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Proof of Theorem idssxp

Step	Hyp	Ref	Expression
1		fnresi 5922	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		fnrel 5903	. . 3 ⊢ (( I ↾ 𝐴) Fn 𝐴 → Rel ( I ↾ 𝐴))
3		relssdmrn 5573	. . 3 ⊢ (Rel ( I ↾ 𝐴) → ( I ↾ 𝐴) ⊆ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)))
4	1, 2, 3	mp2b 10	. 2 ⊢ ( I ↾ 𝐴) ⊆ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴))
5		dmresi 5376	. . 3 ⊢ dom ( I ↾ 𝐴) = 𝐴
6		rnresi 5398	. . 3 ⊢ ran ( I ↾ 𝐴) = 𝐴
7	5, 6	xpeq12i 5061	. 2 ⊢ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) = (𝐴 × 𝐴)
8	4, 7	sseqtri 3600	1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Syntax hints:   ⊆ wss 3540   I cid 4948   × cxp 5036  dom cdm 5038  ran crn 5039   ↾ cres 5040  Rel wrel 5043   Fn wfn 5799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807

This theorem is referenced by:  qtophaus  29231