Theorem idssxp 5796

Description: A diagonal set as a subset of a Cartesian product. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof shortened by BJ, 9-Sep-2022.)

Assertion

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

Proof of Theorem idssxp

Step	Hyp	Ref	Expression
1		idinxpres 5795	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
2		inss2 4126	. 2 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
3	1, 2	eqsstrri 3923	1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Syntax hints:   ∩ cin 3858   ⊆ wss 3859   I cid 5347   × cxp 5441   ↾ cres 5445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-res 5455

This theorem is referenced by:  resiexg  7475  hartogslem1  8852  dfle2  12390  hausdiag  21937  qtophaus  30717  idresssidinxp  35117  rtrclex  39481  rtrclexi  39485  idhe  39637