Theorem exmidfodomrlemeldju 7266

Description: Lemma for exmidfodomr 7271. A variant of djur 7135. (Contributed by Jim Kingdon, 2-Jul-2022.)

Hypotheses

Ref	Expression
exmidfodomrlemeldju.a	|- ( ph -> A C_ 1o )
exmidfodomrlemeldju.el	|- ( ph -> B e. ( A ⊔ 1o ) )

Assertion

Ref	Expression
exmidfodomrlemeldju	|- ( ph -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )

Proof of Theorem exmidfodomrlemeldju

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidfodomrlemeldju.a	. . . . . . . . . 10 |- ( ph -> A C_ 1o )
2	1	sselda 3183	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. 1o )
3		el1o 6495	. . . . . . . . 9 |- ( x e. 1o <-> x = (/) )
4	2, 3	sylib 122	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> x = (/) )
5	4	fveq2d 5562	. . . . . . 7 |- ( ( ph /\ x e. A ) -> (inl ` x ) = (inl ` (/) ) )
6	5	eqeq2d 2208	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( B = (inl ` x ) <-> B = (inl ` (/) ) ) )
7	6	biimpd 144	. . . . 5 |- ( ( ph /\ x e. A ) -> ( B = (inl ` x ) -> B = (inl ` (/) ) ) )
8	7	rexlimdva 2614	. . . 4 |- ( ph -> ( E. x e. A B = (inl ` x ) -> B = (inl ` (/) ) ) )
9	8	imp 124	. . 3 |- ( ( ph /\ E. x e. A B = (inl ` x ) ) -> B = (inl ` (/) ) )
10	9	orcd 734	. 2 |- ( ( ph /\ E. x e. A B = (inl ` x ) ) -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
11		simpr 110	. . . . . . . . 9 |- ( ( ph /\ x e. 1o ) -> x e. 1o )
12	11, 3	sylib 122	. . . . . . . 8 |- ( ( ph /\ x e. 1o ) -> x = (/) )
13	12	fveq2d 5562	. . . . . . 7 |- ( ( ph /\ x e. 1o ) -> (inr ` x ) = (inr ` (/) ) )
14	13	eqeq2d 2208	. . . . . 6 |- ( ( ph /\ x e. 1o ) -> ( B = (inr ` x ) <-> B = (inr ` (/) ) ) )
15	14	biimpd 144	. . . . 5 |- ( ( ph /\ x e. 1o ) -> ( B = (inr ` x ) -> B = (inr ` (/) ) ) )
16	15	rexlimdva 2614	. . . 4 |- ( ph -> ( E. x e. 1o B = (inr ` x ) -> B = (inr ` (/) ) ) )
17	16	imp 124	. . 3 |- ( ( ph /\ E. x e. 1o B = (inr ` x ) ) -> B = (inr ` (/) ) )
18	17	olcd 735	. 2 |- ( ( ph /\ E. x e. 1o B = (inr ` x ) ) -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
19		exmidfodomrlemeldju.el	. . 3 |- ( ph -> B e. ( A ⊔ 1o ) )
20		djur 7135	. . 3 |- ( B e. ( A ⊔ 1o ) <-> ( E. x e. A B = (inl ` x ) \/ E. x e. 1o B = (inr ` x ) ) )
21	19, 20	sylib 122	. 2 |- ( ph -> ( E. x e. A B = (inl ` x ) \/ E. x e. 1o B = (inr ` x ) ) )
22	10, 18, 21	mpjaodan 799	1 |- ( ph -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2167   E.wrex 2476   C_ wss 3157   (/)c0 3450   ` cfv 5258   1oc1o 6467   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114

This theorem is referenced by:  exmidfodomrlemr  7269