Theorem idssxpOLD 6242

Description: Obsolete version of idssxp 5697 as of 9-Sep-2022. (Contributed by Thierry Arnoux, 29-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem idssxpOLD

Step	Hyp	Ref	Expression
1		fnresi 6241	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
2		fnrel 6222	. . 3 ⊢ (( I ↾ 𝐴) Fn 𝐴 → Rel ( I ↾ 𝐴))
3		relssdmrn 5897	. . 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 5700	. . 3 ⊢ dom ( I ↾ 𝐴) = 𝐴
6		rnresi 5720	. . 3 ⊢ ran ( I ↾ 𝐴) = 𝐴
7	5, 6	xpeq12i 5370	. 2 ⊢ (dom ( I ↾ 𝐴) × ran ( I ↾ 𝐴)) = (𝐴 × 𝐴)
8	4, 7	sseqtri 3862	1 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Syntax hints:   ⊆ wss 3798   I cid 5249   × cxp 5340  dom cdm 5342  ran crn 5343   ↾ cres 5344  Rel wrel 5347   Fn wfn 6118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-fun 6125  df-fn 6126

This theorem is referenced by: (None)