Theorem fodjuomni 7316

Description: A condition which ensures A is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)

Hypotheses

Ref	Expression
fodjuomni.o	|- ( ph -> O e. Omni )
fodjuomni.fo	|- ( ph -> F : O -onto-> ( A ⊔ B ) )

Assertion

Ref	Expression
fodjuomni	|- ( ph -> ( E. x x e. A \/ A = (/) ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)   F( x)   O( x)

Proof of Theorem fodjuomni

Dummy variables a b y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fodjuomni.o	. 2 |- ( ph -> O e. Omni )
2		fodjuomni.fo	. 2 |- ( ph -> F : O -onto-> ( A ⊔ B ) )
3		fveq2 5627	. . . . . . 7 |- ( b = z -> (inl ` b ) = (inl ` z ) )
4	3	eqeq2d 2241	. . . . . 6 |- ( b = z -> ( ( F ` a ) = (inl ` b ) <-> ( F ` a ) = (inl ` z ) ) )
5	4	cbvrexv 2766	. . . . 5 |- ( E. b e. A ( F ` a ) = (inl ` b ) <-> E. z e. A ( F ` a ) = (inl ` z ) )
6		ifbi 3623	. . . . 5 |- ( ( E. b e. A ( F ` a ) = (inl ` b ) <-> E. z e. A ( F ` a ) = (inl ` z ) ) -> if ( E. b e. A ( F ` a ) = (inl ` b ) , (/) , 1o ) = if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o ) )
7	5, 6	ax-mp 5	. . . 4 |- if ( E. b e. A ( F ` a ) = (inl ` b ) , (/) , 1o ) = if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o )
8	7	mpteq2i 4171	. . 3 |- ( a e. O |-> if ( E. b e. A ( F ` a ) = (inl ` b ) , (/) , 1o ) ) = ( a e. O |-> if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o ) )
9		fveq2 5627	. . . . . . 7 |- ( a = y -> ( F ` a ) = ( F ` y ) )
10	9	eqeq1d 2238	. . . . . 6 |- ( a = y -> ( ( F ` a ) = (inl ` z ) <-> ( F ` y ) = (inl ` z ) ) )
11	10	rexbidv 2531	. . . . 5 |- ( a = y -> ( E. z e. A ( F ` a ) = (inl ` z ) <-> E. z e. A ( F ` y ) = (inl ` z ) ) )
12	11	ifbid 3624	. . . 4 |- ( a = y -> if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o ) = if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
13	12	cbvmptv 4180	. . 3 |- ( a e. O |-> if ( E. z e. A ( F ` a ) = (inl ` z ) , (/) , 1o ) ) = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
14	8, 13	eqtri 2250	. 2 |- ( a e. O |-> if ( E. b e. A ( F ` a ) = (inl ` b ) , (/) , 1o ) ) = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )
15	1, 2, 14	fodjuomnilemres 7315	1 |- ( ph -> ( E. x x e. A \/ A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 713   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   (/)c0 3491   ifcif 3602   |-> cmpt 4145   -onto->wfo 5316   ` cfv 5318   1oc1o 6555   ⊔ cdju 7204  inlcinl 7212  Omnicomni 7301

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  ax-setind 4629

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-rab 2517  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-ima 4732  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-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-1o 6562  df-2o 6563  df-map 6797  df-dju 7205  df-inl 7214  df-inr 7215  df-omni 7302

This theorem is referenced by:  ctssexmid  7317  exmidunben  12997  exmidsbthrlem  16390  sbthomlem  16393