Theorem funimacnv 3571

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 3570	. . . 4 |- (Fun F -> `'(`'F |` A) = (F |` (`'F"A)))
2	1	rneqd 3341	. . 3 |- (Fun F -> ran `'(`'F |` A) = ran ( F |` (`'F"A)))
3		df-ima 3191	. . 3 |- (F"(`'F"A)) = ran ( F |` (`'F"A))
4	2, 3	syl6reqr 1526	. 2 |- (Fun F -> (F"(`'F"A)) = ran `'(`'F |` A))
5		df-rn 3189	. . . 4 |- ran F = dom `' F
6	5	ineq2i 2214	. . 3 |- (A i^i ran F) = (A i^i dom `' F)
7		dmres 3380	. . 3 |- dom (`'F |` A) = (A i^i dom `' F)
8		dfdm4 3305	. . 3 |- dom (`'F |` A) = ran `'(`'F |` A)
9	6, 7, 8	3eqtr2r 1502	. 2 |- ran `'(`'F |` A) = (A i^i ran F)
10	4, 9	syl6eq 1523	1 |- (Fun F -> (F"(`'F"A)) = (A i^i ran F))

Syntax hints:   -> wi 3   = wceq 956   i^i cin 2046  `'ccnv 3169  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173  Fun wfun 3176

This theorem is referenced by:  funimass1 3572  funimass2 3573  cnsscnp 7772

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-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

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