Theorem snopsuppss 8204

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 8202	. 2 ⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ dom {⟨𝑋, 𝑌⟩}
2		dmsnopss 6234	. 2 ⊢ dom {⟨𝑋, 𝑌⟩} ⊆ {𝑋}
3	1, 2	sstri 3993	1 ⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ {𝑋}

Syntax hints:   ⊆ wss 3951  {csn 4626  ⟨cop 4632  dom cdm 5685  (class class class)co 7431   supp csupp 8185

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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186

This theorem is referenced by:  snopfsupp  9431