Theorem exmidfodomrlemeldju 7072

Description: Lemma for exmidfodomr 7077. A variant of djur 6962. (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 3102	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. 1o )
3		el1o 6342	. . . . . . . . 9 |- ( x e. 1o <-> x = (/) )
4	2, 3	sylib 121	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> x = (/) )
5	4	fveq2d 5433	. . . . . . 7 |- ( ( ph /\ x e. A ) -> (inl ` x ) = (inl ` (/) ) )
6	5	eqeq2d 2152	. . . . . 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 2552	. . . 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 723	. 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 5433	. . . . . . 7 |- ( ( ph /\ x e. 1o ) -> (inr ` x ) = (inr ` (/) ) )
14	13	eqeq2d 2152	. . . . . 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 2552	. . . 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 724	. 2 |- ( ( ph /\ E. x e. 1o B = (inr ` x ) ) -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
19		exmidfodomrlemeldju.el	. . 3 |- ( ph -> B e. ( A ⊔ 1o ) )
20		djur 6962	. . 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 788	1 |- ( ph -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1332   e. wcel 1481   E.wrex 2418   C_ wss 3076   (/)c0 3368   ` cfv 5131   1oc1o 6314   ⊔ cdju 6930  inlcinl 6938  inrcinr 6939

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-dju 6931  df-inl 6940  df-inr 6941

This theorem is referenced by:  exmidfodomrlemr  7075