Theorem fimacnvdisj 4590

Description: The pre-image of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)

Assertion

Ref	Expression
fimacnvdisj	|- ((F:A-->B /\ (B i^i C) = (/)) -> (`'F"C) = (/))

Proof of Theorem fimacnvdisj

Step	Hyp	Ref	Expression
1		frn 4569	. . . . 5 |- (F:A-->B -> ran F C_ B)
2	1	adantr 425	. . . 4 |- ((F:A-->B /\ (B i^i C) = (/)) -> ran F C_ B)
3		df-rn 4005	. . . 4 |- ran F = dom `' F
4	2, 3	syl5ssr 2662	. . 3 |- ((F:A-->B /\ (B i^i C) = (/)) -> dom `' F C_ B)
5		ssdisj 2923	. . 3 |- ((dom `' F C_ B /\ (B i^i C) = (/)) -> (dom `' F i^i C) = (/))
6	4, 5	sylancom 531	. 2 |- ((F:A-->B /\ (B i^i C) = (/)) -> (dom `' F i^i C) = (/))
7		imadisj 4285	. 2 |- ((`'F"C) = (/) <-> (dom `' F i^i C) = (/))
8	6, 7	sylibr 217	1 |- ((F:A-->B /\ (B i^i C) = (/)) -> (`'F"C) = (/))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   i^i cin 2592   C_ wss 2593  (/)c0 2875  `'ccnv 3985  dom cdm 3986  ran crn 3987  "cima 3989  -->wf 3994

This theorem is referenced by:  cnconst 9057

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-f 4010

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