Theorem foelcdmi 5630

Description: A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.)

Assertion

Ref	Expression
foelcdmi	|- ( ( F : A -onto-> B /\ Y e. B ) -> E. x e. A ( F ` x ) = Y )

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

Proof of Theorem foelcdmi

Step	Hyp	Ref	Expression
1		forn 5500	. . . 4 |- ( F : A -onto-> B -> ran F = B )
2	1	eleq2d 2274	. . 3 |- ( F : A -onto-> B -> ( Y e. ran F <-> Y e. B ) )
3		fofn 5499	. . . 4 |- ( F : A -onto-> B -> F Fn A )
4		fvelrnb 5625	. . . 4 |- ( F Fn A -> ( Y e. ran F <-> E. x e. A ( F ` x ) = Y ) )
5	3, 4	syl 14	. . 3 |- ( F : A -onto-> B -> ( Y e. ran F <-> E. x e. A ( F ` x ) = Y ) )
6	2, 5	bitr3d 190	. 2 |- ( F : A -onto-> B -> ( Y e. B <-> E. x e. A ( F ` x ) = Y ) )
7	6	biimpa 296	1 |- ( ( F : A -onto-> B /\ Y e. B ) -> E. x e. A ( F ` x ) = Y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   E.wrex 2484   ran crn 4675   Fn wfn 5265   -onto->wfo 5268   ` cfv 5270

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fo 5276  df-fv 5278

This theorem is referenced by:  mhmid  13422  mhmmnd  13423  ghmgrp  13425  ghmcmn  13634  imasabl  13643