Theorem fimacnv 3816

Description: The pre-image 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 3421	. . . 4 |- (`'F"B) (_ ran `' F
2	1	a1i 8	. . 3 |- (F:A-->B -> (`'F"B) (_ ran `' F)
3		fdm 3637	. . . . 5 |- (F:A-->B -> dom F = A)
4		ssid 2083	. . . . . 6 |- A (_ A
5	4	a1i 8	. . . . 5 |- (F:A-->B -> A (_ A)
6	3, 5	eqsstrd 2098	. . . 4 |- (F:A-->B -> dom F (_ A)
7		dfdm4 3311	. . . 4 |- dom F = ran `' F
8	6, 7	syl5ssr 2109	. . 3 |- (F:A-->B -> ran `' F (_ A)
9	2, 8	sstrd 2077	. 2 |- (F:A-->B -> (`'F"B) (_ A)
10		imassrn 3421	. . . . 5 |- (F"A) (_ ran F
11	10	a1i 8	. . . 4 |- (F:A-->B -> (F"A) (_ ran F)
12		frn 3639	. . . 4 |- (F:A-->B -> ran F (_ B)
13	11, 12	sstrd 2077	. . 3 |- (F:A-->B -> (F"A) (_ B)
14		funimass3 3812	. . . 4 |- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A (_ (`'F"B)))
15		ffun 3635	. . . 4 |- (F:A-->B -> Fun F)
16	5, 3	sseqtr4d 2101	. . . 4 |- (F:A-->B -> A (_ dom F)
17	14, 15, 16	sylanc 473	. . 3 |- (F:A-->B -> ((F"A) (_ B <-> A (_ (`'F"B)))
18	13, 17	mpbid 195	. 2 |- (F:A-->B -> A (_ (`'F"B))
19	9, 18	eqssd 2082	1 |- (F:A-->B -> (`'F"B) = A)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   (_ wss 2050  `'ccnv 3175  dom cdm 3176  ran crn 3177  "cima 3179  Fun wfun 3182  -->wf 3184

This theorem is referenced by:  iscncl 7767  mapudiscn 10498  eqindhome 10527

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204

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