Theorem fodjum 7313

Description: Lemma for fodjuomni 7316 and fodjumkv 7327. 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 6578	. . . . . . . . 9 |- 1o =/= (/)
3	2	nesymi 2446	. . . . . . . 8 |- -. (/) = 1o
4	3	intnan 934	. . . . . . 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 5636	. . . . . . . . . . 11 |- ( y = w -> ( ( F ` y ) = (inl ` z ) <-> ( F ` w ) = (inl ` z ) ) )
9	8	rexbidv 2531	. . . . . . . . . 10 |- ( y = w -> ( E. z e. A ( F ` y ) = (inl ` z ) <-> E. z e. A ( F ` w ) = (inl ` z ) ) )
10	9	ifbid 3624	. . . . . . . . 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 4686	. . . . . . . . . . 11 |- (/) e. om
13	12	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) e. om )
14		1onn 6666	. . . . . . . . . . 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 7311	. . . . . . . . . . 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 3640	. . . . . . . . 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 5728	. . . . . . . 8 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( P ` w ) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
21	6, 20	eqtr3d 2264	. . . . . . 7 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
22		eqifdc 3639	. . . . . . . 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 1383	. . . . 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 3591	. . . 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 2290	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
30	29	cbvexv 1965	. . 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 2653	1 |- ( ph -> E. x x e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   (/)c0 3491   ifcif 3602   |-> cmpt 4145   omcom 4682   -onto->wfo 5316   ` cfv 5318   1oc1o 6555   ⊔ cdju 7204  inlcinl 7212

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6286  df-2nd 6287  df-1o 6562  df-dju 7205  df-inl 7214  df-inr 7215

This theorem is referenced by:  fodjuomnilemres  7315  fodjumkvlemres  7326