Theorem cnvss 5817

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 5119	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
2	1	ssopab2dv 5496	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
3		df-cnv 5629	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
4		df-cnv 5629	. 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 5075  {copab 5137  ^◡ccnv 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ss 3902  df-br 5076  df-opab 5138  df-cnv 5629

This theorem is referenced by:  cnveq  5818  rnss  5888  relcnvtrg  6222  predrelss  6292  funss  6508  funres11  6566  funcnvres  6567  foimacnv  6788  funcnvuni  7876  tposss  8171  vdwnnlem1  16961  structcnvcnv  17118  catcoppccl  18079  cnvps  18539  tsrdir  18565  ustneism  24211  metustsym  24542  metust  24545  pi1xfrcnv  25046  eulerpartlemmf  34571  relcnveq3  38709  elrelscnveq3  39009  disjss  39213  cnvssb  44045  trclubgNEW  44077  clrellem  44081  clcnvlem  44082  cnvrcl0  44084  cnvtrcl0  44085  cnvtrrel  44129  relexpaddss  44177