Theorem f1ocnvfv3 6077

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 5990	. . 3 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( `' F ` C ) e. A )
2		f1ocnvfvb 5989	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ x e. A /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
3	2	3expa 1234	. . . . 5 |- ( ( ( F : A -1-1-onto-> B /\ x e. A ) /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
4	3	an32s 574	. . . 4 |- ( ( ( F : A -1-1-onto-> B /\ C e. B ) /\ x e. A ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
5		eqcom 2240	. . . 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 6069	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( iota_ x e. A ( F ` x ) = C ) = ( `' F ` C ) )
8	7	eqcomd 2244	1 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( `' F ` C ) = ( iota_ x e. A ( F ` x ) = C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   `'ccnv 4775   -1-1-onto->wf1o 5379   ` cfv 5380   iota_crio 6040

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041

This theorem is used by: (None)