Theorem cnvss 5746

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 5113	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
2	1	ssopab2dv 5441	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
3		df-cnv 5566	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
4		df-cnv 5566	. 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
5	2, 3, 4	3sstr4g 4015	1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Syntax hints:   → wi 4   ⊆ wss 3939   class class class wbr 5069  {copab 5131  ^◡ccnv 5557

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-in 3946  df-ss 3955  df-br 5070  df-opab 5132  df-cnv 5566

This theorem is referenced by:  cnveq  5747  rnss  5812  relcnvtrg  6122  funss  6377  funres11  6434  funcnvres  6435  foimacnv  6635  funcnvuni  7639  tposss  7896  vdwnnlem1  16334  structcnvcnv  16500  catcoppccl  17371  cnvps  17825  tsrdir  17851  ustneism  22835  metustsym  23168  metust  23171  pi1xfrcnv  23664  eulerpartlemmf  31637  relcnveq3  35582  elrelscnveq3  35735  disjss  35968  cnvssb  39952  trclubgNEW  39984  clrellem  39988  clcnvlem  39989  cnvrcl0  39991  cnvtrcl0  39992  cnvtrrel  40021  relexpaddss  40069