Theorem exmidfodomrlemeldju 7048

Description: Lemma for exmidfodomr 7053. A variant of djur 6947. (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 3092	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. 1o )
3		el1o 6327	. . . . . . . . 9 |- ( x e. 1o <-> x = (/) )
4	2, 3	sylib 121	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> x = (/) )
5	4	fveq2d 5418	. . . . . . 7 |- ( ( ph /\ x e. A ) -> (inl ` x ) = (inl ` (/) ) )
6	5	eqeq2d 2149	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( B = (inl ` x ) <-> B = (inl ` (/) ) ) )
7	6	biimpd 143	. . . . 5 |- ( ( ph /\ x e. A ) -> ( B = (inl ` x ) -> B = (inl ` (/) ) ) )
8	7	rexlimdva 2547	. . . 4 |- ( ph -> ( E. x e. A B = (inl ` x ) -> B = (inl ` (/) ) ) )
9	8	imp 123	. . 3 |- ( ( ph /\ E. x e. A B = (inl ` x ) ) -> B = (inl ` (/) ) )
10	9	orcd 722	. 2 |- ( ( ph /\ E. x e. A B = (inl ` x ) ) -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
11		simpr 109	. . . . . . . . 9 |- ( ( ph /\ x e. 1o ) -> x e. 1o )
12	11, 3	sylib 121	. . . . . . . 8 |- ( ( ph /\ x e. 1o ) -> x = (/) )
13	12	fveq2d 5418	. . . . . . 7 |- ( ( ph /\ x e. 1o ) -> (inr ` x ) = (inr ` (/) ) )
14	13	eqeq2d 2149	. . . . . 6 |- ( ( ph /\ x e. 1o ) -> ( B = (inr ` x ) <-> B = (inr ` (/) ) ) )
15	14	biimpd 143	. . . . 5 |- ( ( ph /\ x e. 1o ) -> ( B = (inr ` x ) -> B = (inr ` (/) ) ) )
16	15	rexlimdva 2547	. . . 4 |- ( ph -> ( E. x e. 1o B = (inr ` x ) -> B = (inr ` (/) ) ) )
17	16	imp 123	. . 3 |- ( ( ph /\ E. x e. 1o B = (inr ` x ) ) -> B = (inr ` (/) ) )
18	17	olcd 723	. 2 |- ( ( ph /\ E. x e. 1o B = (inr ` x ) ) -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
19		exmidfodomrlemeldju.el	. . 3 |- ( ph -> B e. ( A ⊔ 1o ) )
20		djur 6947	. . 3 |- ( B e. ( A ⊔ 1o ) <-> ( E. x e. A B = (inl ` x ) \/ E. x e. 1o B = (inr ` x ) ) )
21	19, 20	sylib 121	. 2 |- ( ph -> ( E. x e. A B = (inl ` x ) \/ E. x e. 1o B = (inr ` x ) ) )
22	10, 18, 21	mpjaodan 787	1 |- ( ph -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   E.wrex 2415   C_ wss 3066   (/)c0 3358   ` cfv 5118   1oc1o 6299   ⊔ cdju 6915  inlcinl 6923  inrcinr 6924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-dju 6916  df-inl 6925  df-inr 6926

This theorem is referenced by:  exmidfodomrlemr  7051