Theorem fdmeu 5743

Description: There is exactly one codomain element for each element of the domain of a function. (Contributed by AV, 20-Apr-2025.)

Assertion

Ref	Expression
fdmeu	|- ( ( F : A --> B /\ X e. A ) -> E! y e. B ( F ` X ) = y )

Distinct variable groups:   y, A   y, B   y, F   y, X

Proof of Theorem fdmeu

Step	Hyp	Ref	Expression
1		feu 5572	. 2 |- ( ( F : A --> B /\ X e. A ) -> E! y e. B <. X , y >. e. F )
2		ffn 5531	. . . . . 6 |- ( F : A --> B -> F Fn A )
3	2	anim1i 340	. . . . 5 |- ( ( F : A --> B /\ X e. A ) -> ( F Fn A /\ X e. A ) )
4	3	adantr 276	. . . 4 |- ( ( ( F : A --> B /\ X e. A ) /\ y e. B ) -> ( F Fn A /\ X e. A ) )
5		fnopfvb 5739	. . . 4 |- ( ( F Fn A /\ X e. A ) -> ( ( F ` X ) = y <-> <. X , y >. e. F ) )
6	4, 5	syl 14	. . 3 |- ( ( ( F : A --> B /\ X e. A ) /\ y e. B ) -> ( ( F ` X ) = y <-> <. X , y >. e. F ) )
7	6	reubidva 2736	. 2 |- ( ( F : A --> B /\ X e. A ) -> ( E! y e. B ( F ` X ) = y <-> E! y e. B <. X , y >. e. F ) )
8	1, 7	mpbird 167	1 |- ( ( F : A --> B /\ X e. A ) -> E! y e. B ( F ` X ) = y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E!wreu 2530   <.cop 3711   Fn wfn 5370   -->wf 5371   ` cfv 5375

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 4247  ax-pow 4309  ax-pr 4344

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-reu 2535  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383

This theorem is referenced by:  uspgriedgedg  16403