Metamath Proof Explorer


Theorem fvimacnv

Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006)

Ref Expression
Assertion fvimacnv Fun F A dom F F A B A F -1 B

Proof

Step Hyp Ref Expression
1 funfvop Fun F A dom F A F A F
2 fvex F A V
3 opelcnvg F A V A dom F F A A F -1 A F A F
4 2 3 mpan A dom F F A A F -1 A F A F
5 4 adantl Fun F A dom F F A A F -1 A F A F
6 1 5 mpbird Fun F A dom F F A A F -1
7 elimasng F A V A dom F A F -1 F A F A A F -1
8 2 7 mpan A dom F A F -1 F A F A A F -1
9 8 adantl Fun F A dom F A F -1 F A F A A F -1
10 6 9 mpbird Fun F A dom F A F -1 F A
11 2 snss F A B F A B
12 imass2 F A B F -1 F A F -1 B
13 11 12 sylbi F A B F -1 F A F -1 B
14 13 sseld F A B A F -1 F A A F -1 B
15 10 14 syl5com Fun F A dom F F A B A F -1 B
16 fvimacnvi Fun F A F -1 B F A B
17 16 ex Fun F A F -1 B F A B
18 17 adantr Fun F A dom F A F -1 B F A B
19 15 18 impbid Fun F A dom F F A B A F -1 B