Theorem fvimacnvALT 3806

Description: Another proof of fvimacnv 3802, based on funimass3 3803. If funimass3 3803 is ever proved directly, as opposed to using funimacnv 3568 pointwise, then the proof of funimacnv 3568 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 3803	. . 3 |- ((Fun F /\ {A} (_ dom F) -> ((F"{A}) (_ B <-> {A} (_ (`'F"B)))
2		snssi 2464	. . 3 |- (A e. dom F -> {A} (_ dom F)
3	1, 2	sylan2 451	. 2 |- ((Fun F /\ A e. dom F) -> ((F"{A}) (_ B <-> {A} (_ (`'F"B)))
4		fnsnfv 3764	. . . . 5 |- ((F Fn dom F /\ A e. dom F) -> {(F` A)} = (F"{A}))
5		eqid 1475	. . . . . 6 |- dom F = dom F
6		df-fn 3190	. . . . . . 7 |- (F Fn dom F <-> (Fun F /\ dom F = dom F))
7	6	biimpr 152	. . . . . 6 |- ((Fun F /\ dom F = dom F) -> F Fn dom F)
8	5, 7	mpan2 695	. . . . 5 |- (Fun F -> F Fn dom F)
9	4, 8	sylan 448	. . . 4 |- ((Fun F /\ A e. dom F) -> {(F` A)} = (F"{A}))
10	9	sseq1d 2086	. . 3 |- ((Fun F /\ A e. dom F) -> ({(F` A)} (_ B <-> (F"{A}) (_ B))
11		fvex 3729	. . . 4 |- (F` A) e. V
12	11	snss 2459	. . 3 |- ((F` A) e. B <-> {(F` A)} (_ B)
13	10, 12	syl5bb 531	. 2 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> (F"{A}) (_ B))
14		snssg 2461	. . 3 |- (A e. dom F -> (A e. (`'F"B) <-> {A} (_ (`'F"B)))
15	14	adantl 388	. 2 |- ((Fun F /\ A e. dom F) -> (A e. (`'F"B) <-> {A} (_ (`'F"B)))
16	3, 13, 15	3bitr4d 549	1 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957   (_ wss 2045  {csn 2407  `'ccnv 3166  dom cdm 3167  "cima 3170  Fun wfun 3173   Fn wfn 3174  ` cfv 3179

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-fv 3195

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