Theorem snopsuppss 8220

Description: The support of a singleton containing an ordered pair is a subset of the singleton containing the first element of the ordered pair, i.e. it is empty or the singleton itself. (Contributed by AV, 19-Jul-2019.)

Assertion

Ref	Expression
snopsuppss	⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ {𝑋}

Proof of Theorem snopsuppss

Step	Hyp	Ref	Expression
1		suppssdm 8218	. 2 ⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ dom {⟨𝑋, 𝑌⟩}
2		dmsnopss 6245	. 2 ⊢ dom {⟨𝑋, 𝑌⟩} ⊆ {𝑋}
3	1, 2	sstri 4018	1 ⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ {𝑋}

Syntax hints:   ⊆ wss 3976  {csn 4648  ⟨cop 4654  dom cdm 5700  (class class class)co 7448   supp csupp 8201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-supp 8202

This theorem is referenced by:  snopfsupp  9460