Theorem snopsuppss 8158

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 8156	. 2 ⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ dom {⟨𝑋, 𝑌⟩}
2		dmsnopss 6203	. 2 ⊢ dom {⟨𝑋, 𝑌⟩} ⊆ {𝑋}
3	1, 2	sstri 3983	1 ⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ {𝑋}

Syntax hints:   ⊆ wss 3940  {csn 4620  ⟨cop 4626  dom cdm 5666  (class class class)co 7401   supp csupp 8140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-supp 8141

This theorem is referenced by:  snopfsupp  9382