Theorem fimacnv 5625

Description: The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

Assertion

Ref	Expression
fimacnv	|- ( F : A --> B -> ( `' F " B ) = A )

Proof of Theorem fimacnv

Step	Hyp	Ref	Expression
1		imassrn 4964	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4803	. . . 4 |- dom F = ran `' F
3		fdm 5353	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3167	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3199	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3194	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3158	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 4964	. . . 4 |- ( F " A ) C_ ran F
9		frn 5356	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3158	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5350	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3198	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5612	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
14	11, 12, 13	syl2anc 409	. . 3 |- ( F : A --> B -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
15	10, 14	mpbid 146	. 2 |- ( F : A --> B -> A C_ ( `' F " B ) )
16	7, 15	eqssd 3164	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   C_ wss 3121   `'ccnv 4610   dom cdm 4611   ran crn 4612   "cima 4614   Fun wfun 5192   -->wf 5194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206

This theorem is referenced by:  fmpt  5646  nn0supp  9187  cnclima  13017