Theorem fodju0 7477

Description: Lemma for fodjuomni 7479 and fodjumkv 7490. 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 7381	. . . . 5 |- ( u e. A -> (inl ` u ) e. ( A ⊔ B ) )
3		foelrn 5948	. . . . 5 |- ( ( F : O -onto-> ( A ⊔ B ) /\ (inl ` u ) e. ( A ⊔ B ) ) -> E. v e. O (inl ` u ) = ( F ` v ) )
4	1, 2, 3	syl2an 289	. . . 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 5699	. . . . . . . 8 |- ( y = v -> ( ( F ` y ) = (inl ` z ) <-> ( F ` v ) = (inl ` z ) ) )
7	6	rexbidv 2551	. . . . . . 7 |- ( y = v -> ( E. z e. A ( F ` y ) = (inl ` z ) <-> E. z e. A ( F ` v ) = (inl ` z ) ) )
8	7	ifbid 3659	. . . . . 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 535	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> v e. O )
10		peano1 4736	. . . . . . . 8 |- (/) e. om
11	10	a1i 9	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (/) e. om )
12		1onn 6783	. . . . . . . 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 7474	. . . . . . . 8 |- ( ( ph /\ v e. O ) -> DECID E. z e. A ( F ` v ) = (inl ` z ) )
15	14	ad2ant2r 513	. . . . . . 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 3675	. . . . . 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 5793	. . . . 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 5699	. . . . . 6 |- ( w = v -> ( ( P ` w ) = 1o <-> ( P ` v ) = 1o ) )
19		fodju0.1	. . . . . . 7 |- ( ph -> A. w e. O ( P ` w ) = 1o )
20	19	ad2antrr 492	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> A. w e. O ( P ` w ) = 1o )
21	18, 20, 9	rspcdva 2934	. . . . 5 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( P ` v ) = 1o )
22		simplr 533	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> u e. A )
23		simprr 537	. . . . . . . 8 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (inl ` u ) = ( F ` v ) )
24	23	eqcomd 2244	. . . . . . 7 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> ( F ` v ) = (inl ` u ) )
25		fveq2 5690	. . . . . . . 8 |- ( z = u -> (inl ` z ) = (inl ` u ) )
26	25	rspceeqv 2948	. . . . . . 7 |- ( ( u e. A /\ ( F ` v ) = (inl ` u ) ) -> E. z e. A ( F ` v ) = (inl ` z ) )
27	22, 24, 26	syl2anc 415	. . . . . 6 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> E. z e. A ( F ` v ) = (inl ` z ) )
28	27	iftrued 3644	. . . . 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 2280	. . . 4 |- ( ( ( ph /\ u e. A ) /\ ( v e. O /\ (inl ` u ) = ( F ` v ) ) ) -> (/) = 1o )
30	4, 29	rexlimddv 2673	. . 3 |- ( ( ph /\ u e. A ) -> (/) = 1o )
31		1n0 6695	. . . . 5 |- 1o =/= (/)
32	31	nesymi 2466	. . . 4 |- -. (/) = 1o
33	32	a1i 9	. . 3 |- ( ( ph /\ u e. A ) -> -. (/) = 1o )
34	30, 33	pm2.65da 671	. 2 |- ( ph -> -. u e. A )
35	34	eq0rdv 3570	1 |- ( ph -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   (/)c0 3520   ifcif 3635   |-> cmpt 4187   omcom 4732   -onto->wfo 5370   ` cfv 5372   1oc1o 6670   ⊔ cdju 7367  inlcinl 7375

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-in1 623  ax-in2 624  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-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-dju 7368  df-inl 7377  df-inr 7378

This theorem is referenced by:  fodjuomnilemres  7478  fodjumkvlemres  7489