Theorem fodjuomni 7149

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 5517	. . . . . . 7 |- ( b = z -> (inl ` b ) = (inl ` z ) )
4	3	eqeq2d 2189	. . . . . 6 |- ( b = z -> ( ( F ` a ) = (inl ` b ) <-> ( F ` a ) = (inl ` z ) ) )
5	4	cbvrexv 2706	. . . . 5 |- ( E. b e. A ( F ` a ) = (inl ` b ) <-> E. z e. A ( F ` a ) = (inl ` z ) )
6		ifbi 3556	. . . . 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 4092	. . 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 5517	. . . . . . 7 |- ( a = y -> ( F ` a ) = ( F ` y ) )
10	9	eqeq1d 2186	. . . . . 6 |- ( a = y -> ( ( F ` a ) = (inl ` z ) <-> ( F ` y ) = (inl ` z ) ) )
11	10	rexbidv 2478	. . . . 5 |- ( a = y -> ( E. z e. A ( F ` a ) = (inl ` z ) <-> E. z e. A ( F ` y ) = (inl ` z ) ) )
12	11	ifbid 3557	. . . 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 4101	. . 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 2198	. 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 7148	1 |- ( ph -> ( E. x x e. A \/ A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 708   = wceq 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   (/)c0 3424   ifcif 3536   |-> cmpt 4066   -onto->wfo 5216   ` cfv 5218   1oc1o 6412   ⊔ cdju 7038  inlcinl 7046  Omnicomni 7134

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-1o 6419  df-2o 6420  df-map 6652  df-dju 7039  df-inl 7048  df-inr 7049  df-omni 7135

This theorem is referenced by:  ctssexmid  7150  exmidunben  12429  exmidsbthrlem  14855  sbthomlem  14858