Theorem fvimacnvALT 4782

Description: Another proof of fvimacnv 4778, based on funimass3 4779. If funimass3 4779 is ever proved directly, as opposed to using funimacnv 4490 pointwise, then the proof of funimacnv 4490 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
fvimacnvALT	|- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))

Proof of Theorem fvimacnvALT

Step	Hyp	Ref	Expression
1		funimass3 4779	. . 3 |- ((Fun F /\ {A} C_ dom F) -> ((F"{A}) C_ B <-> {A} C_ (`'F"B)))
2		snssi 3129	. . 3 |- (A e. dom F -> {A} C_ dom F)
3	1, 2	sylan2 500	. 2 |- ((Fun F /\ A e. dom F) -> ((F"{A}) C_ B <-> {A} C_ (`'F"B)))
4		fnsnfv 4728	. . . . 5 |- ((F Fn dom F /\ A e. dom F) -> {(F` A)} = (F"{A}))
5		eqid 1884	. . . . . 6 |- dom F = dom F
6		df-fn 4009	. . . . . . 7 |- (F Fn dom F <-> (Fun F /\ dom F = dom F))
7	6	biimpri 169	. . . . . 6 |- ((Fun F /\ dom F = dom F) -> F Fn dom F)
8	5, 7	mpan2 760	. . . . 5 |- (Fun F -> F Fn dom F)
9	4, 8	sylan 497	. . . 4 |- ((Fun F /\ A e. dom F) -> {(F` A)} = (F"{A}))
10	9	sseq1d 2644	. . 3 |- ((Fun F /\ A e. dom F) -> ({(F` A)} C_ B <-> (F"{A}) C_ B))
11		fvex 4689	. . . 4 |- (F` A) e. _V
12	11	snss 3122	. . 3 |- ((F` A) e. B <-> {(F` A)} C_ B)
13	10, 12	syl5bb 591	. 2 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> (F"{A}) C_ B))
14		snssg 3124	. . 3 |- (A e. dom F -> (A e. (`'F"B) <-> {A} C_ (`'F"B)))
15	14	adantl 424	. 2 |- ((Fun F /\ A e. dom F) -> (A e. (`'F"B) <-> {A} C_ (`'F"B)))
16	3, 13, 15	3bitr4d 609	1 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  {csn 3044  `'ccnv 3985  dom cdm 3986  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

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-13 1311  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  ax-un 3790

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-ral 2109  df-rex 2110  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-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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