Theorem fimacnv 5711

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 5034	. . 3 |- ( `' F " B ) C_ ran `' F
2		dfdm4 4871	. . . 4 |- dom F = ran `' F
3		fdm 5433	. . . . 5 |- ( F : A --> B -> dom F = A )
4		ssid 3213	. . . . 5 |- A C_ A
5	3, 4	eqsstrdi 3245	. . . 4 |- ( F : A --> B -> dom F C_ A )
6	2, 5	eqsstrrid 3240	. . 3 |- ( F : A --> B -> ran `' F C_ A )
7	1, 6	sstrid 3204	. 2 |- ( F : A --> B -> ( `' F " B ) C_ A )
8		imassrn 5034	. . . 4 |- ( F " A ) C_ ran F
9		frn 5436	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	8, 9	sstrid 3204	. . 3 |- ( F : A --> B -> ( F " A ) C_ B )
11		ffun 5430	. . . 4 |- ( F : A --> B -> Fun F )
12	4, 3	sseqtrrid 3244	. . . 4 |- ( F : A --> B -> A C_ dom F )
13		funimass3 5698	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
14	11, 12, 13	syl2anc 411	. . 3 |- ( F : A --> B -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )
15	10, 14	mpbid 147	. 2 |- ( F : A --> B -> A C_ ( `' F " B ) )
16	7, 15	eqssd 3210	1 |- ( F : A --> B -> ( `' F " B ) = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   C_ wss 3166   `'ccnv 4675   dom cdm 4676   ran crn 4677   "cima 4679   Fun wfun 5266   -->wf 5268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280

This theorem is referenced by:  fmpt  5732  nn0supp  9349  cnclima  14728