Theorem snopsuppss 7995

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 7993	. 2 ⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ dom {⟨𝑋, 𝑌⟩}
2		dmsnopss 6117	. 2 ⊢ dom {⟨𝑋, 𝑌⟩} ⊆ {𝑋}
3	1, 2	sstri 3930	1 ⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ {𝑋}

Syntax hints:   ⊆ wss 3887  {csn 4561  ⟨cop 4567  dom cdm 5589  (class class class)co 7275   supp csupp 7977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-supp 7978

This theorem is referenced by:  snopfsupp  9151