Theorem funss 5127

Description: Subclass theorem for function predicate. (The proof was shortened by Mario Carneiro, 24-Jun-2014.) (Contributed by set.mm contributors, 16-Aug-1994.) (Revised by set.mm contributors, 24-Jun-2014.)

Assertion

Ref	Expression
funss	⊢ (A ⊆ B → (Fun B → Fun A))

Proof of Theorem funss

Step	Hyp	Ref	Expression
1		coss1 4873	. . . 4 ⊢ (A ⊆ B → (A ∘ ◡A) ⊆ (B ∘ ◡A))
2		cnvss 4886	. . . . 5 ⊢ (A ⊆ B → ◡A ⊆ ◡B)
3		coss2 4874	. . . . 5 ⊢ (◡A ⊆ ◡B → (B ∘ ◡A) ⊆ (B ∘ ◡B))
4	2, 3	syl 15	. . . 4 ⊢ (A ⊆ B → (B ∘ ◡A) ⊆ (B ∘ ◡B))
5	1, 4	sstrd 3283	. . 3 ⊢ (A ⊆ B → (A ∘ ◡A) ⊆ (B ∘ ◡B))
6		sstr2 3280	. . 3 ⊢ ((A ∘ ◡A) ⊆ (B ∘ ◡B) → ((B ∘ ◡B) ⊆ I → (A ∘ ◡A) ⊆ I ))
7	5, 6	syl 15	. 2 ⊢ (A ⊆ B → ((B ∘ ◡B) ⊆ I → (A ∘ ◡A) ⊆ I ))
8		df-fun 4790	. 2 ⊢ (Fun B ↔ (B ∘ ◡B) ⊆ I )
9		df-fun 4790	. 2 ⊢ (Fun A ↔ (A ∘ ◡A) ⊆ I )
10	7, 8, 9	3imtr4g 261	1 ⊢ (A ⊆ B → (Fun B → Fun A))

Syntax hints:   → wi 4   ⊆ wss 3258   ∘ ccom 4722   I cid 4764  ^◡ccnv 4772  Fun wfun 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727  df-cnv 4786  df-fun 4790

This theorem is referenced by:  funeq  5128  funopab4  5142  funres  5144  funin  5164  funres11  5165  foimacnv  5304