Theorem f1ocnvfv3 5880

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 5798	. . 3 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( `' F ` C ) e. A )
2		f1ocnvfvb 5797	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ x e. A /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
3	2	3expa 1205	. . . . 5 |- ( ( ( F : A -1-1-onto-> B /\ x e. A ) /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
4	3	an32s 568	. . . 4 |- ( ( ( F : A -1-1-onto-> B /\ C e. B ) /\ x e. A ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
5		eqcom 2191	. . . 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 5872	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( iota_ x e. A ( F ` x ) = C ) = ( `' F ` C ) )
8	7	eqcomd 2195	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 1364   e. wcel 2160   `'ccnv 4640   -1-1-onto->wf1o 5230   ` cfv 5231   iota_crio 5846

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847

This theorem is referenced by: (None)