Theorem foelrni 6244

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
foelrni	|- ( ( 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 foelrni

Step	Hyp	Ref	Expression
1		forn 6118	. . . 4 |- ( F : A -onto-> B -> ran F = B )
2	1	eleq2d 2687	. . 3 |- ( F : A -onto-> B -> ( Y e. ran F <-> Y e. B ) )
3		fofn 6117	. . . 4 |- ( F : A -onto-> B -> F Fn A )
4		fvelrnb 6243	. . . 4 |- ( F Fn A -> ( Y e. ran F <-> E. x e. A ( F ` x ) = Y ) )
5	3, 4	syl 17	. . 3 |- ( F : A -onto-> B -> ( Y e. ran F <-> E. x e. A ( F ` x ) = Y ) )
6	2, 5	bitr3d 270	. 2 |- ( F : A -onto-> B -> ( Y e. B <-> E. x e. A ( F ` x ) = Y ) )
7	6	biimpa 501	1 |- ( ( F : A -onto-> B /\ Y e. B ) -> E. x e. A ( F ` x ) = Y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   ran crn 5115   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896

This theorem is referenced by:  mhmid  17536  mhmmnd  17537  ghmgrp  17539  symgmov2  17813  ghmcmn  18237  founiiun  39360  founiiun0  39377  sge0f1o  40599  isomenndlem  40744  ovnsubaddlem1  40784