Theorem funimadisj 3606

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 3587	. . . . 5 |- (F Fn A -> dom F = A)
2	1	ineq1d 2216	. . . 4 |- (F Fn A -> (dom F i^i C) = (A i^i C))
3	2	eqeq1d 1483	. . 3 |- (F Fn A -> ((dom F i^i C) = (/) <-> (A i^i C) = (/)))
4	3	biimpar 417	. 2 |- ((F Fn A /\ (A i^i C) = (/)) -> (dom F i^i C) = (/))
5		imadisj 3422	. 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 956   i^i cin 2046  (/)c0 2280  dom cdm 3170  "cima 3173   Fn wfn 3177

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

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-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-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fn 3193

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