Theorem fimacnv 3810

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 3415	. . . 4 |- (`'F"B) (_ ran `' F
2	1	a1i 8	. . 3 |- (F:A-->B -> (`'F"B) (_ ran `' F)
3		fdm 3631	. . . . 5 |- (F:A-->B -> dom F = A)
4		ssid 2080	. . . . . 6 |- A (_ A
5	4	a1i 8	. . . . 5 |- (F:A-->B -> A (_ A)
6	3, 5	eqsstrd 2095	. . . 4 |- (F:A-->B -> dom F (_ A)
7		dfdm4 3305	. . . 4 |- dom F = ran `' F
8	6, 7	syl5ssr 2106	. . 3 |- (F:A-->B -> ran `' F (_ A)
9	2, 8	sstrd 2074	. 2 |- (F:A-->B -> (`'F"B) (_ A)
10		imassrn 3415	. . . . 5 |- (F"A) (_ ran F
11	10	a1i 8	. . . 4 |- (F:A-->B -> (F"A) (_ ran F)
12		frn 3633	. . . 4 |- (F:A-->B -> ran F (_ B)
13	11, 12	sstrd 2074	. . 3 |- (F:A-->B -> (F"A) (_ B)
14		funimass3 3806	. . . 4 |- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A (_ (`'F"B)))
15		ffun 3629	. . . 4 |- (F:A-->B -> Fun F)
16	5, 3	sseqtr4d 2098	. . . 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 2079	1 |- (F:A-->B -> (`'F"B) = A)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   (_ wss 2047  `'ccnv 3169  dom cdm 3170  ran crn 3171  "cima 3173  Fun wfun 3176  -->wf 3178

This theorem is referenced by:  iscncl 7770  mapudiscn 10512  eqindhome 10541

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198

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