Theorem foelrnf 39373

Description: Property of a surjective function. As foelrn 6378 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
foelrnf.1	|- F/_ x F

Assertion

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

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

Allowed substitution hint:   F( x)

Proof of Theorem foelrnf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		foelrnf.1	. . . 4 |- F/_ x F
2	1	dffo3f 39364	. . 3 |- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )
3	2	simprbi 480	. 2 |- ( F : A -onto-> B -> A. y e. B E. x e. A y = ( F ` x ) )
4		eqeq1 2626	. . . 4 |- ( y = C -> ( y = ( F ` x ) <-> C = ( F ` x ) ) )
5	4	rexbidv 3052	. . 3 |- ( y = C -> ( E. x e. A y = ( F ` x ) <-> E. x e. A C = ( F ` x ) ) )
6	5	rspccva 3308	. 2 |- ( ( A. y e. B E. x e. A y = ( F ` x ) /\ C e. B ) -> E. x e. A C = ( F ` x ) )
7	3, 6	sylan 488	1 |- ( ( F : A -onto-> B /\ C e. B ) -> E. x e. A C = ( F ` x ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   E.wrex 2913   -->wf 5884   -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:  fompt  39379