Theorem funimacnv 3577

Description: The image of the pre-image of a function.

Assertion

Ref	Expression
funimacnv	|- (Fun F -> (F"(`'F"A)) = (A i^i ran F))

Proof of Theorem funimacnv

Step	Hyp	Ref	Expression
1		funcnvres2 3576	. . . 4 |- (Fun F -> `'(`'F |` A) = (F |` (`'F"A)))
2	1	rneqd 3347	. . 3 |- (Fun F -> ran `'(`'F |` A) = ran ( F |` (`'F"A)))
3		df-ima 3197	. . 3 |- (F"(`'F"A)) = ran ( F |` (`'F"A))
4	2, 3	syl6reqr 1529	. 2 |- (Fun F -> (F"(`'F"A)) = ran `'(`'F |` A))
5		df-rn 3195	. . . 4 |- ran F = dom `' F
6	5	ineq2i 2217	. . 3 |- (A i^i ran F) = (A i^i dom `' F)
7		dmres 3386	. . 3 |- dom (`'F |` A) = (A i^i dom `' F)
8		dfdm4 3311	. . 3 |- dom (`'F |` A) = ran `'(`'F |` A)
9	6, 7, 8	3eqtr2r 1505	. 2 |- ran `'(`'F |` A) = (A i^i ran F)
10	4, 9	syl6eq 1526	1 |- (Fun F -> (F"(`'F"A)) = (A i^i ran F))

Syntax hints:   -> wi 3   = wceq 958   i^i cin 2049  `'ccnv 3175  dom cdm 3176  ran crn 3177   |` cres 3178  "cima 3179  Fun wfun 3182

This theorem is referenced by:  funimass1 3578  funimass2 3579  cnsscnp 7769

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-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198

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