Theorem fodju0 7019

Description: Lemma for fodjuomni 7021 and fodjumkv 7034. A condition which shows that A is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)

Hypotheses

Ref	Expression
fodjuf.fo	|- ( ph -> F : O -onto-> ( A ⊔ B ) )
fodjuf.p	|- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
fodju0.1	|- ( ph -> A. w e. O ( P ` w ) = 1o )

Assertion

Ref	Expression
fodju0	|- ( ph -> A = (/) )

Distinct variable groups:   ph, y, z   y, O, z   z, A   z, B   z, F   y, A   y, F   w, O   w, P

Allowed substitution hints:   ph( w)   A( w)   B( y, w)   P( y, z)   F( w)

Proof of Theorem fodju0

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjuf.fo	. . . . 5 |- ( ph -> F : O -onto-> ( A ⊔ B ) )
2		djulcl 6936	. . . . 5 |- ( u e. A -> (inl ` u ) e. ( A ⊔ B ) )
3		foelrn 5654	. . . . 5 |- ( ( F : O -onto-> ( A ⊔ B ) /\ (inl ` u ) e. ( A ⊔ B ) ) -> E. v e. O (inl ` u ) = ( F ` v ) )
4	1, 2, 3	syl2an 287	. . . 4 |- ( ( ph /\ u e. A ) -> E. v e. O (inl ` u ) = ( F ` v ) )
5		fodjuf.p	. . . . . 6 |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
6		fveqeq2 5430	. . . . . . . 8 |- ( y = v -> ( ( F ` y ) = (inl ` z ) <-> ( F ` v ) = (inl ` z ) ) )
7	6	rexbidv 2438	. . . . . . 7 |- ( y = v -> ( E. z e. A ( F ` y ) = (inl ` z ) <-> E. z e. A ( F ` v ) = (inl ` z ) ) )
8	7	ifbid 3493	. . . . . 6 |- ( y = v -> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) = if ( E. z e. A ( F ` v ) = (inl ` z ) , (/) , 1o ) )
9		simprl 520	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> v e. O )
10		peano1 4508	. . . . . . . 8 |- (/) e. om
11	10	a1i 9	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (/) e. om )
12		1onn 6416	. . . . . . . 8 |- 1o e. om
13	12	a1i 9	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> 1o e. om )
14	1	fodjuomnilemdc 7016	. . . . . . . 8 |- ( ( ph /\ v e. O ) -> DECID E. z e. A ( F ` v ) = (inl ` z ) )
15	14	ad2ant2r 500	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> DECID E. z e. A ( F ` v ) = (inl ` z ) )
16	11, 13, 15	ifcldcd 3507	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> if ( E. z e. A ( F ` v ) = (inl ` z ) , (/) , 1o ) e. om )
17	5, 8, 9, 16	fvmptd3 5514	. . . . 5 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( P ` v ) = if ( E. z e. A ( F ` v ) = (inl ` z ) , (/) , 1o ) )
18		fveqeq2 5430	. . . . . 6 |- ( w = v -> ( ( P ` w ) = 1o <-> ( P ` v ) = 1o ) )
19		fodju0.1	. . . . . . 7 |- ( ph -> A. w e. O ( P ` w ) = 1o )
20	19	ad2antrr 479	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> A. w e. O ( P ` w ) = 1o )
21	18, 20, 9	rspcdva 2794	. . . . 5 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( P ` v ) = 1o )
22		simplr 519	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> u e. A )
23		simprr 521	. . . . . . . 8 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (inl ` u ) = ( F ` v ) )
24	23	eqcomd 2145	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( F ` v ) = (inl ` u ) )
25		fveq2 5421	. . . . . . . 8 |- ( z = u -> (inl ` z ) = (inl ` u ) )
26	25	rspceeqv 2807	. . . . . . 7 |- ( ( u e. A /\ ( F ` v ) = (inl ` u ) ) -> E. z e. A ( F ` v ) = (inl ` z ) )
27	22, 24, 26	syl2anc 408	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> E. z e. A ( F ` v ) = (inl ` z ) )
28	27	iftrued 3481	. . . . 5 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> if ( E. z e. A ( F ` v ) = (inl ` z ) , (/) , 1o ) = (/) )
29	17, 21, 28	3eqtr3rd 2181	. . . 4 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (/) = 1o )
30	4, 29	rexlimddv 2554	. . 3 |- ( ( ph /\ u e. A ) -> (/) = 1o )
31		1n0 6329	. . . . 5 |- 1o =/= (/)
32	31	nesymi 2354	. . . 4 |- -. (/) = 1o
33	32	a1i 9	. . 3 |- ( ( ph /\ u e. A ) -> -. (/) = 1o )
34	30, 33	pm2.65da 650	. 2 |- ( ph -> -. u e. A )
35	34	eq0rdv 3407	1 |- ( ph -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103  DECID wdc 819   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   (/)c0 3363   ifcif 3474   |-> cmpt 3989   omcom 4504   -onto->wfo 5121   ` cfv 5123   1oc1o 6306   ⊔ cdju 6922  inlcinl 6930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933

This theorem is referenced by:  fodjuomnilemres  7020  fodjumkvlemres  7033