Theorem cnvss 5816

Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.)

Assertion

Ref	Expression
cnvss	⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Proof of Theorem cnvss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssbr 5118	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
2	1	ssopab2dv 5495	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
3		df-cnv 5628	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
4		df-cnv 5628	. 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
5	2, 3, 4	3sstr4g 3970	1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Syntax hints:   → wi 4   ⊆ wss 3885   class class class wbr 5074  {copab 5136  ^◡ccnv 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ss 3902  df-br 5075  df-opab 5137  df-cnv 5628

This theorem is referenced by:  cnveq  5817  rnss  5883  relcnvtrg  6220  predrelss  6290  funss  6506  funres11  6564  funcnvres  6565  foimacnv  6786  funcnvuni  7872  tposss  8166  vdwnnlem1  16955  structcnvcnv  17112  catcoppccl  18073  cnvps  18533  tsrdir  18559  ustneism  24177  metustsym  24508  metust  24511  pi1xfrcnv  25012  eulerpartlemmf  34507  relcnveq3  38636  elrelscnveq3  38936  disjss  39140  cnvssb  44001  trclubgNEW  44033  clrellem  44037  clcnvlem  44038  cnvrcl0  44040  cnvtrcl0  44041  cnvtrrel  44085  relexpaddss  44133