Theorem foelcdmi 5749

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 5613	. . . 4 |- ( F : A -onto-> B -> ran F = B )
2	1	eleq2d 2308	. . 3 |- ( F : A -onto-> B -> ( Y e. ran F <-> Y e. B ) )
3		fofn 5612	. . . 4 |- ( F : A -onto-> B -> F Fn A )
4		fvelrnb 5744	. . . 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 1402   e. wcel 2209   E.wrex 2529   ran crn 4770   Fn wfn 5367   -onto->wfo 5370   ` cfv 5372

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 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 4244  ax-pow 4306  ax-pr 4341

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380

This theorem is referenced by:  mhmid  13895  mhmmnd  13896  ghmgrp  13898  ghmcmn  14108  imasabl  14117