Theorem dffo3f 39364

Description: An onto mapping expressed in terms of function values. As dffo3 6374 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
dffo3f.1	|- F/_ x F

Assertion

Ref	Expression
dffo3f	|- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )

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

Allowed substitution hint:   F( x)

Proof of Theorem dffo3f

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffo2 6119	. 2 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
2		ffn 6045	. . . . 5 |- ( F : A --> B -> F Fn A )
3		fnrnfv 6242	. . . . . . 7 |- ( F Fn A -> ran F = { y | E. w e. A y = ( F ` w ) } )
4		nfcv 2764	. . . . . . . . . . 11 |- F/_ x y
5		dffo3f.1	. . . . . . . . . . . 12 |- F/_ x F
6		nfcv 2764	. . . . . . . . . . . 12 |- F/_ x w
7	5, 6	nffv 6198	. . . . . . . . . . 11 |- F/_ x ( F ` w )
8	4, 7	nfeq 2776	. . . . . . . . . 10 |- F/ x y = ( F ` w )
9		nfv 1843	. . . . . . . . . 10 |- F/ w y = ( F ` x )
10		fveq2 6191	. . . . . . . . . . 11 |- ( w = x -> ( F ` w ) = ( F ` x ) )
11	10	eqeq2d 2632	. . . . . . . . . 10 |- ( w = x -> ( y = ( F ` w ) <-> y = ( F ` x ) ) )
12	8, 9, 11	cbvrex 3168	. . . . . . . . 9 |- ( E. w e. A y = ( F ` w ) <-> E. x e. A y = ( F ` x ) )
13	12	abbii 2739	. . . . . . . 8 |- { y | E. w e. A y = ( F ` w ) } = { y | E. x e. A y = ( F ` x ) }
14	13	a1i 11	. . . . . . 7 |- ( F Fn A -> { y | E. w e. A y = ( F ` w ) } = { y | E. x e. A y = ( F ` x ) } )
15	3, 14	eqtrd 2656	. . . . . 6 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
16	15	eqeq1d 2624	. . . . 5 |- ( F Fn A -> ( ran F = B <-> { y | E. x e. A y = ( F ` x ) } = B ) )
17	2, 16	syl 17	. . . 4 |- ( F : A --> B -> ( ran F = B <-> { y | E. x e. A y = ( F ` x ) } = B ) )
18		nfcv 2764	. . . . . . . . . 10 |- F/_ x A
19		nfcv 2764	. . . . . . . . . 10 |- F/_ x B
20	5, 18, 19	nff 6041	. . . . . . . . 9 |- F/ x F : A --> B
21		nfv 1843	. . . . . . . . 9 |- F/ x y e. B
22		simpr 477	. . . . . . . . . . 11 |- ( ( ( F : A --> B /\ x e. A ) /\ y = ( F ` x ) ) -> y = ( F ` x ) )
23		ffvelrn 6357	. . . . . . . . . . . 12 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
24	23	adantr 481	. . . . . . . . . . 11 |- ( ( ( F : A --> B /\ x e. A ) /\ y = ( F ` x ) ) -> ( F ` x ) e. B )
25	22, 24	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ( F : A --> B /\ x e. A ) /\ y = ( F ` x ) ) -> y e. B )
26	25	exp31 630	. . . . . . . . 9 |- ( F : A --> B -> ( x e. A -> ( y = ( F ` x ) -> y e. B ) ) )
27	20, 21, 26	rexlimd 3026	. . . . . . . 8 |- ( F : A --> B -> ( E. x e. A y = ( F ` x ) -> y e. B ) )
28	27	biantrurd 529	. . . . . . 7 |- ( F : A --> B -> ( ( y e. B -> E. x e. A y = ( F ` x ) ) <-> ( ( E. x e. A y = ( F ` x ) -> y e. B ) /\ ( y e. B -> E. x e. A y = ( F ` x ) ) ) ) )
29		dfbi2 660	. . . . . . 7 |- ( ( E. x e. A y = ( F ` x ) <-> y e. B ) <-> ( ( E. x e. A y = ( F ` x ) -> y e. B ) /\ ( y e. B -> E. x e. A y = ( F ` x ) ) ) )
30	28, 29	syl6rbbr 279	. . . . . 6 |- ( F : A --> B -> ( ( E. x e. A y = ( F ` x ) <-> y e. B ) <-> ( y e. B -> E. x e. A y = ( F ` x ) ) ) )
31	30	albidv 1849	. . . . 5 |- ( F : A --> B -> ( A. y ( E. x e. A y = ( F ` x ) <-> y e. B ) <-> A. y ( y e. B -> E. x e. A y = ( F ` x ) ) ) )
32		abeq1 2733	. . . . 5 |- ( { y | E. x e. A y = ( F ` x ) } = B <-> A. y ( E. x e. A y = ( F ` x ) <-> y e. B ) )
33		df-ral 2917	. . . . 5 |- ( A. y e. B E. x e. A y = ( F ` x ) <-> A. y ( y e. B -> E. x e. A y = ( F ` x ) ) )
34	31, 32, 33	3bitr4g 303	. . . 4 |- ( F : A --> B -> ( { y | E. x e. A y = ( F ` x ) } = B <-> A. y e. B E. x e. A y = ( F ` x ) ) )
35	17, 34	bitrd 268	. . 3 |- ( F : A --> B -> ( ran F = B <-> A. y e. B E. x e. A y = ( F ` x ) ) )
36	35	pm5.32i 669	. 2 |- ( ( F : A --> B /\ ran F = B ) <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )
37	1, 36	bitri 264	1 |- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   A.wral 2912   E.wrex 2913   ran crn 5115   Fn wfn 5883   -->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:  foelrnf  39373  fompt  39379