Theorem cbvexfo 5937

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 5570	. . 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 2234	. . . 4 |- ( y = ( F ` x ) -> ( ps <-> ph ) )
5	4	rexrn 5792	. . 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 5571	. . 3 |- ( F : A -onto-> B -> ran F = B )
8	7	rexeqdv 2738	. 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 ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   E.wrex 2512   ran crn 4732   Fn wfn 5328   -onto->wfo 5331   ` cfv 5333

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341

This theorem is referenced by: (None)