Theorem cnvss 5827

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 5146	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
2	1	ssopab2dv 5506	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
3		df-cnv 5639	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
4		df-cnv 5639	. 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
5	2, 3, 4	3sstr4g 3997	1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Syntax hints:   → wi 4   ⊆ wss 3911   class class class wbr 5102  {copab 5164  ^◡ccnv 5630

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ss 3928  df-br 5103  df-opab 5165  df-cnv 5639

This theorem is referenced by:  cnveq  5828  rnss  5893  relcnvtrg  6228  predrelss  6299  funss  6520  funres11  6578  funcnvres  6579  foimacnv  6800  funcnvuni  7889  tposss  8184  vdwnnlem1  16944  structcnvcnv  17101  catcoppccl  18061  cnvps  18521  tsrdir  18547  ustneism  24146  metustsym  24478  metust  24481  pi1xfrcnv  24992  eulerpartlemmf  34361  relcnveq3  38304  elrelscnveq3  38477  disjss  38718  cnvssb  43570  trclubgNEW  43602  clrellem  43606  clcnvlem  43607  cnvrcl0  43609  cnvtrcl0  43610  cnvtrrel  43654  relexpaddss  43702