Theorem fimacnv 3805

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 3411	. . . 4 |- (`'F"B) (_ ran `' F
2	1	a1i 8	. . 3 |- (F:A-->B -> (`'F"B) (_ ran `' F)
3		fdm 3627	. . . . 5 |- (F:A-->B -> dom F = A)
4		ssid 2077	. . . . . 6 |- A (_ A
5	4	a1i 8	. . . . 5 |- (F:A-->B -> A (_ A)
6	3, 5	eqsstrd 2092	. . . 4 |- (F:A-->B -> dom F (_ A)
7		dfdm4 3301	. . . 4 |- dom F = ran `' F
8	6, 7	syl5ssr 2103	. . 3 |- (F:A-->B -> ran `' F (_ A)
9	2, 8	sstrd 2071	. 2 |- (F:A-->B -> (`'F"B) (_ A)
10		imassrn 3411	. . . . 5 |- (F"A) (_ ran F
11	10	a1i 8	. . . 4 |- (F:A-->B -> (F"A) (_ ran F)
12		frn 3628	. . . 4 |- (F:A-->B -> ran F (_ B)
13	11, 12	sstrd 2071	. . 3 |- (F:A-->B -> (F"A) (_ B)
14		funimass3 3801	. . . 4 |- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A (_ (`'F"B)))
15		ffun 3625	. . . 4 |- (F:A-->B -> Fun F)
16	5, 3	sseqtr4d 2095	. . . 4 |- (F:A-->B -> A (_ dom F)
17	14, 15, 16	sylanc 471	. . 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 2076	1 |- (F:A-->B -> (`'F"B) = A)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   (_ wss 2044  `'ccnv 3165  dom cdm 3166  ran crn 3167  "cima 3169  Fun wfun 3172  -->wf 3174

This theorem is referenced by:  iscncl 7730  mapudiscn 10458

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194

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