Theorem fvelrnb 5628

Description: A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)

Assertion

Ref	Expression
fvelrnb	|- ( F Fn A -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) )

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

Proof of Theorem fvelrnb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2490	. . . 4 |- ( E. x e. A ( F ` x ) = B <-> E. x ( x e. A /\ ( F ` x ) = B ) )
2		19.41v 1926	. . . . 5 |- ( E. x ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) <-> ( E. x ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) )
3		simpl 109	. . . . . . . . . 10 |- ( ( x e. A /\ ( F ` x ) = B ) -> x e. A )
4	3	anim1i 340	. . . . . . . . 9 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( x e. A /\ F Fn A ) )
5	4	ancomd 267	. . . . . . . 8 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( F Fn A /\ x e. A ) )
6		funfvex 5595	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
7	6	funfni 5377	. . . . . . . 8 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
8	5, 7	syl 14	. . . . . . 7 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( F ` x ) e. _V )
9		simpr 110	. . . . . . . . 9 |- ( ( x e. A /\ ( F ` x ) = B ) -> ( F ` x ) = B )
10	9	eleq1d 2274	. . . . . . . 8 |- ( ( x e. A /\ ( F ` x ) = B ) -> ( ( F ` x ) e. _V <-> B e. _V ) )
11	10	adantr 276	. . . . . . 7 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( ( F ` x ) e. _V <-> B e. _V ) )
12	8, 11	mpbid 147	. . . . . 6 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
13	12	exlimiv 1621	. . . . 5 |- ( E. x ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
14	2, 13	sylbir 135	. . . 4 |- ( ( E. x ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
15	1, 14	sylanb 284	. . 3 |- ( ( E. x e. A ( F ` x ) = B /\ F Fn A ) -> B e. _V )
16	15	expcom 116	. 2 |- ( F Fn A -> ( E. x e. A ( F ` x ) = B -> B e. _V ) )
17		fnrnfv 5627	. . . 4 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
18	17	eleq2d 2275	. . 3 |- ( F Fn A -> ( B e. ran F <-> B e. { y | E. x e. A y = ( F ` x ) } ) )
19		eqeq1 2212	. . . . . 6 |- ( y = B -> ( y = ( F ` x ) <-> B = ( F ` x ) ) )
20		eqcom 2207	. . . . . 6 |- ( B = ( F ` x ) <-> ( F ` x ) = B )
21	19, 20	bitrdi 196	. . . . 5 |- ( y = B -> ( y = ( F ` x ) <-> ( F ` x ) = B ) )
22	21	rexbidv 2507	. . . 4 |- ( y = B -> ( E. x e. A y = ( F ` x ) <-> E. x e. A ( F ` x ) = B ) )
23	22	elab3g 2924	. . 3 |- ( ( E. x e. A ( F ` x ) = B -> B e. _V ) -> ( B e. { y | E. x e. A y = ( F ` x ) } <-> E. x e. A ( F ` x ) = B ) )
24	18, 23	sylan9bbr 463	. 2 |- ( ( ( E. x e. A ( F ` x ) = B -> B e. _V ) /\ F Fn A ) -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) )
25	16, 24	mpancom 422	1 |- ( F Fn A -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1515   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772   ran crn 4677   Fn wfn 5267   ` cfv 5272

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280

This theorem is referenced by:  foelcdmi  5633  chfnrn  5693  rexrn  5719  ralrn  5720  elrnrexdmb  5722  ffnfv  5740  fconstfvm  5804  elunirn  5837  isoini  5889  canth  5899  reldm  6274  ordiso2  7139  eldju  7172  ctssdc  7217  uzn0  9666  frec2uzrand  10552  frecuzrdgtcl  10559  frecuzrdgfunlem  10566  uzin2  11331  imasmnd2  13317  imasgrp2  13479  imasrng  13751  imasring  13859  reeff1o  15278