Theorem funimass2 2713

Description: A kind of contraposition law that infers an image subclass from a subclass of a converse image.

Assertion

Ref	Expression
funimass2	⊢ (Fun F → (A ⊆ (◡F “ B) → (F “ A) ⊆ B))

Proof of Theorem funimass2

Step	Hyp	Ref	Expression
1		funimacnv 2711	. . . 4 ⊢ (Fun F → (F “ (◡F “ B)) = (B ∩ ran F))
2	1	sseq2d 1528	. . 3 ⊢ (Fun F → ((F “ A) ⊆ (F “ (◡F “ B)) ↔ (F “ A) ⊆ (B ∩ ran F)))
3		inss1 1657	. . . 4 ⊢ (B ∩ ran F) ⊆ B
4		sstr2 1510	. . . 4 ⊢ ((F “ A) ⊆ (B ∩ ran F) → ((B ∩ ran F) ⊆ B → (F “ A) ⊆ B))
5	3, 4	mpi 44	. . 3 ⊢ ((F “ A) ⊆ (B ∩ ran F) → (F “ A) ⊆ B)
6	2, 5	syl6bi 187	. 2 ⊢ (Fun F → ((F “ A) ⊆ (F “ (◡F “ B)) → (F “ A) ⊆ B))
7		imass2 2622	. 2 ⊢ (A ⊆ (◡F “ B) → (F “ A) ⊆ (F “ (◡F “ B)))
8	6, 7	syl5 22	1 ⊢ (Fun F → (A ⊆ (◡F “ B) → (F “ A) ⊆ B))

Syntax hints:   → wi 2   ∩ cin 1486   ⊆ wss 1487  ^◡ccnv 2409  ran crn 2411   “ cima 2413  Fun wfun 2416

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432

metamath.org