Theorem fodjum 7405

Description: Lemma for fodjuomni 7408 and fodjumkv 7419. 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 6643	. . . . . . . . 9 |- 1o =/= (/)
3	2	nesymi 2449	. . . . . . . 8 |- -. (/) = 1o
4	3	intnan 937	. . . . . . 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 533	. . . . . . . 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 5657	. . . . . . . . . . 11 |- ( y = w -> ( ( F ` y ) = (inl ` z ) <-> ( F ` w ) = (inl ` z ) ) )
9	8	rexbidv 2534	. . . . . . . . . 10 |- ( y = w -> ( E. z e. A ( F ` y ) = (inl ` z ) <-> E. z e. A ( F ` w ) = (inl ` z ) ) )
10	9	ifbid 3631	. . . . . . . . 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 531	. . . . . . . . 9 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> w e. O )
12		peano1 4698	. . . . . . . . . . 11 |- (/) e. om
13	12	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) e. om )
14		1onn 6731	. . . . . . . . . . 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 7403	. . . . . . . . . . 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 3647	. . . . . . . . 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 5749	. . . . . . . 8 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> ( P ` w ) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
21	6, 20	eqtr3d 2266	. . . . . . 7 |- ( ( ph /\ ( w e. O /\ ( P ` w ) = (/) ) ) -> (/) = if ( E. z e. A ( F ` w ) = (inl ` z ) , (/) , 1o ) )
22		eqifdc 3646	. . . . . . . 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 1386	. . . . 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 3596	. . . 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 2292	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
30	29	cbvexv 1967	. . 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 2656	1 |- ( ph -> E. x x e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   E.wex 1541   e. wcel 2202   E.wrex 2512   (/)c0 3496   ifcif 3607   |-> cmpt 4155   omcom 4694   -onto->wfo 5331   ` cfv 5333   1oc1o 6618   ⊔ cdju 7296  inlcinl 7304

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-dju 7297  df-inl 7306  df-inr 7307

This theorem is referenced by:  fodjuomnilemres  7407  fodjumkvlemres  7418