Theorem cbvexfo 5992

Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)

Hypothesis

Ref	Expression
cbvfo.1	|- ( ( F ` x ) = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvexfo	|- ( F : A -onto-> B -> ( E. x e. A ph <-> E. y e. B ps ) )

Distinct variable groups:   x, y, A   y, B   x, F, y   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)   B( x)

Proof of Theorem cbvexfo

Step	Hyp	Ref	Expression
1		fofn 5617	. . 3 |- ( F : A -onto-> B -> F Fn A )
2		cbvfo.1	. . . . . 6 |- ( ( F ` x ) = y -> ( ph <-> ps ) )
3	2	bicomd 141	. . . . 5 |- ( ( F ` x ) = y -> ( ps <-> ph ) )
4	3	eqcoms 2241	. . . 4 |- ( y = ( F ` x ) -> ( ps <-> ph ) )
5	4	rexrn 5845	. . 3 |- ( F Fn A -> ( E. y e. ran F ps <-> E. x e. A ph ) )
6	1, 5	syl 14	. 2 |- ( F : A -onto-> B -> ( E. y e. ran F ps <-> E. x e. A ph ) )
7		forn 5618	. . 3 |- ( F : A -onto-> B -> ran F = B )
8	7	rexeqdv 2756	. 2 |- ( F : A -onto-> B -> ( E. y e. ran F ps <-> E. y e. B ps ) )
9	6, 8	bitr3d 190	1 |- ( F : A -onto-> B -> ( E. x e. A ph <-> E. y e. B ps ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   E.wrex 2529   ran crn 4775   Fn wfn 5372   -onto->wfo 5375   ` cfv 5377

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 4249  ax-pow 4311  ax-pr 4346

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-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 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385

This theorem is used by: (None)