Theorem f1ocnvfv3 6039

Description: Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
f1ocnvfv3	|- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( `' F ` C ) = ( iota_ x e. A ( F ` x ) = C ) )

Distinct variable groups:   x, A   x, B   x, C   x, F

Proof of Theorem f1ocnvfv3

Step	Hyp	Ref	Expression
1		f1ocnvdm 5954	. . 3 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( `' F ` C ) e. A )
2		f1ocnvfvb 5953	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ x e. A /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
3	2	3expa 1230	. . . . 5 |- ( ( ( F : A -1-1-onto-> B /\ x e. A ) /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
4	3	an32s 570	. . . 4 |- ( ( ( F : A -1-1-onto-> B /\ C e. B ) /\ x e. A ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
5		eqcom 2234	. . . 4 |- ( x = ( `' F ` C ) <-> ( `' F ` C ) = x )
6	4, 5	bitr4di 198	. . 3 |- ( ( ( F : A -1-1-onto-> B /\ C e. B ) /\ x e. A ) -> ( ( F ` x ) = C <-> x = ( `' F ` C ) ) )
7	1, 6	riota5 6031	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( iota_ x e. A ( F ` x ) = C ) = ( `' F ` C ) )
8	7	eqcomd 2238	1 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( `' F ` C ) = ( iota_ x e. A ( F ` x ) = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   `'ccnv 4748   -1-1-onto->wf1o 5351   ` cfv 5352   iota_crio 6002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003

This theorem is referenced by: (None)