Theorem fodjum 7250

Description: Lemma for fodjuomni 7253 and fodjumkv 7264. A condition which shows that A is inhabited. (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 ) )
fodjum.z	|- ( ph -> E. w e. O ( P ` w ) = (/) )

Assertion

Ref	Expression
fodjum	|- ( ph -> E. x x e. A )

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

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

Proof of Theorem fodjum

Step	Hyp	Ref	Expression
1		fodjum.z	. 2 |- ( ph -> E. w e. O ( P ` w ) = (/) )
2		1n0 6520	. . . . . . . . 9 |- 1o =/= (/)
3	2	nesymi 2422	. . . . . . . 8 |- -. (/) = 1o
4	3	intnan 931	. . . . . . 7 |- -. ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o )
5	4	a1i 9	. . . . . 6 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> -. ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o ) )
6		simprr 531	. . . . . . . 8 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( P ` w ) = (/) )
7		fodjuf.p	. . . . . . . . 9 |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
8		fveqeq2 5587	. . . . . . . . . . 11 |- ( y = w -> ( ( F ` y ) = (inl ` z ) <-> ( F ` w ) = (inl ` z ) ) )
9	8	rexbidv 2507	. . . . . . . . . 10 |- ( y = w -> ( E. z e. A ( F ` y ) = (inl ` z ) <-> E. z e. A ( F ` w ) = (inl ` z ) ) )
10	9	ifbid 3592	. . . . . . . . 9 |- ( y = w -> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
11		simprl 529	. . . . . . . . 9 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> w e. O )
12		peano1 4643	. . . . . . . . . . 11 |- (/) e. om
13	12	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) e. om )
14		1onn 6608	. . . . . . . . . . 11 |- 1o e. om
15	14	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> 1o e. om )
16		fodjuf.fo	. . . . . . . . . . . 12 |- ( ph -> F : O -onto-> ( A ⊔ B ) )
17	16	fodjuomnilemdc 7248	. . . . . . . . . . 11 |- ( ( ph /\ w e. O ) -> DECID E. z e. A ( F ` w ) = (inl ` z ) )
18	17	adantrr 479	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> DECID E. z e. A ( F ` w ) = (inl ` z ) )
19	13, 15, 18	ifcldcd 3608	. . . . . . . . 9 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) e. om )
20	7, 10, 11, 19	fvmptd3 5675	. . . . . . . 8 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( P ` w ) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
21	6, 20	eqtr3d 2240	. . . . . . 7 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
22		eqifdc 3607	. . . . . . . 8 |- (DECID E. z e. A ( F ` w ) = (inl ` z ) -> ( (/) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) <-> ( ( E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = (/) ) \/ ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o ) ) ) )
23	18, 22	syl 14	. . . . . . 7 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( (/) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) <-> ( ( E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = (/) ) \/ ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o ) ) ) )
24	21, 23	mpbid 147	. . . . . 6 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( ( E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = (/) ) \/ ( -. E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = 1o ) ) )
25	5, 24	ecased 1362	. . . . 5 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( E. z e. A ( F ` w ) = (inl ` z ) /\ (/) = (/) ) )
26	25	simpld 112	. . . 4 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> E. z e. A ( F ` w ) = (inl ` z ) )
27		rexm 3560	. . . 4 |- ( E. z e. A ( F ` w ) = (inl ` z ) -> E. z z e. A )
28	26, 27	syl 14	. . 3 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> E. z z e. A )
29		eleq1w 2266	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
30	29	cbvexv 1942	. . 3 |- ( E. z z e. A <-> E. x x e. A )
31	28, 30	sylib 122	. 2 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> E. x x e. A )
32	1, 31	rexlimddv 2628	1 |- ( ph -> E. x x e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710  DECID wdc 836   = wceq 1373   E.wex 1515   e. wcel 2176   E.wrex 2485   (/)c0 3460   ifcif 3571   |-> cmpt 4106   omcom 4639   -onto->wfo 5270   ` cfv 5272   1oc1o 6497   ⊔ cdju 7141  inlcinl 7149

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-1st 6228  df-2nd 6229  df-1o 6504  df-dju 7142  df-inl 7151  df-inr 7152

This theorem is referenced by:  fodjuomnilemres  7252  fodjumkvlemres  7263