Theorem funimadisj 3612

Description: A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)

Assertion

Ref	Expression
funimadisj	|- ((F Fn A /\ (A i^i C) = (/)) -> (F"C) = (/))

Proof of Theorem funimadisj

Step	Hyp	Ref	Expression
1		fndm 3593	. . . . 5 |- (F Fn A -> dom F = A)
2	1	ineq1d 2219	. . . 4 |- (F Fn A -> (dom F i^i C) = (A i^i C))
3	2	eqeq1d 1486	. . 3 |- (F Fn A -> ((dom F i^i C) = (/) <-> (A i^i C) = (/)))
4	3	biimpar 419	. 2 |- ((F Fn A /\ (A i^i C) = (/)) -> (dom F i^i C) = (/))
5		imadisj 3428	. 2 |- ((F"C) = (/) <-> (dom F i^i C) = (/))
6	4, 5	sylibr 200	1 |- ((F Fn A /\ (A i^i C) = (/)) -> (F"C) = (/))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   i^i cin 2049  (/)c0 2283  dom cdm 3176  "cima 3179   Fn wfn 3183

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

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-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-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fn 3199

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