Theorem fvimacnvALT 3809

Description: Another proof of fvimacnv 3805, based on funimass3 3806. If funimass3 3806 is ever proved directly, as opposed to using funimacnv 3571 pointwise, then the proof of funimacnv 3571 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 3806	. . 3 |- ((Fun F /\ {A} (_ dom F) -> ((F"{A}) (_ B <-> {A} (_ (`'F"B)))
2		snssi 2466	. . 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 3767	. . . . 5 |- ((F Fn dom F /\ A e. dom F) -> {(F` A)} = (F"{A}))
5		eqid 1475	. . . . . 6 |- dom F = dom F
6		df-fn 3193	. . . . . . 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 696	. . . . 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 2088	. . 3 |- ((Fun F /\ A e. dom F) -> ({(F` A)} (_ B <-> (F"{A}) (_ B))
11		fvex 3732	. . . 4 |- (F` A) e. V
12	11	snss 2461	. . 3 |- ((F` A) e. B <-> {(F` A)} (_ B)
13	10, 12	syl5bb 532	. 2 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> (F"{A}) (_ B))
14		snssg 2463	. . 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 550	1 |- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958   (_ wss 2047  {csn 2409  `'ccnv 3169  dom cdm 3170  "cima 3173  Fun wfun 3176   Fn wfn 3177  ` cfv 3182

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  ax-un 2866

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-ral 1649  df-rex 1650  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-uni 2504  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  df-fn 3193  df-fv 3198

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