Theorem exmidfodomrlemreseldju 6816

Description: Lemma for exmidfodomrlemrALT 6819. A variant of eldju 6749. (Contributed by Jim Kingdon, 9-Jul-2022.)

Hypotheses

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

Assertion

Ref	Expression
exmidfodomrlemreseldju	|- ( ph -> ( ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) \/ B = ( (inr |` 1o ) ` (/) ) ) )

Proof of Theorem exmidfodomrlemreseldju

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidfodomrlemreseldju.a	. . . . . . . . . . 11 |- ( ph -> A C_ 1o )
2	1	sselda 3025	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. 1o )
3		el1o 6193	. . . . . . . . . 10 |- ( x e. 1o <-> x = (/) )
4	2, 3	sylib 120	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x = (/) )
5	4	fveq2d 5303	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( (inl |` A ) ` x ) = ( (inl |` A ) ` (/) ) )
6	5	eqeq2d 2099	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( B = ( (inl |` A ) ` x ) <-> B = ( (inl |` A ) ` (/) ) ) )
7		simpr 108	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. A )
8	4, 7	eqeltrrd 2165	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> (/) e. A )
9	8	biantrurd 299	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( B = ( (inl |` A ) ` (/) ) <-> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) ) )
10	6, 9	bitrd 186	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( B = ( (inl |` A ) ` x ) <-> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) ) )
11	10	biimpd 142	. . . . 5 |- ( ( ph /\ x e. A ) -> ( B = ( (inl |` A ) ` x ) -> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) ) )
12	11	rexlimdva 2489	. . . 4 |- ( ph -> ( E. x e. A B = ( (inl |` A ) ` x ) -> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) ) )
13	12	imp 122	. . 3 |- ( ( ph /\ E. x e. A B = ( (inl |` A ) ` x ) ) -> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) )
14	13	orcd 687	. 2 |- ( ( ph /\ E. x e. A B = ( (inl |` A ) ` x ) ) -> ( ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) \/ B = ( (inr |` 1o ) ` (/) ) ) )
15		simpr 108	. . . . . . . . 9 |- ( ( ph /\ x e. 1o ) -> x e. 1o )
16	15, 3	sylib 120	. . . . . . . 8 |- ( ( ph /\ x e. 1o ) -> x = (/) )
17	16	fveq2d 5303	. . . . . . 7 |- ( ( ph /\ x e. 1o ) -> ( (inr |` 1o ) ` x ) = ( (inr |` 1o ) ` (/) ) )
18	17	eqeq2d 2099	. . . . . 6 |- ( ( ph /\ x e. 1o ) -> ( B = ( (inr |` 1o ) ` x ) <-> B = ( (inr |` 1o ) ` (/) ) ) )
19	18	biimpd 142	. . . . 5 |- ( ( ph /\ x e. 1o ) -> ( B = ( (inr |` 1o ) ` x ) -> B = ( (inr |` 1o ) ` (/) ) ) )
20	19	rexlimdva 2489	. . . 4 |- ( ph -> ( E. x e. 1o B = ( (inr |` 1o ) ` x ) -> B = ( (inr |` 1o ) ` (/) ) ) )
21	20	imp 122	. . 3 |- ( ( ph /\ E. x e. 1o B = ( (inr |` 1o ) ` x ) ) -> B = ( (inr |` 1o ) ` (/) ) )
22	21	olcd 688	. 2 |- ( ( ph /\ E. x e. 1o B = ( (inr |` 1o ) ` x ) ) -> ( ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) \/ B = ( (inr |` 1o ) ` (/) ) ) )
23		exmidfodomrlemreseldju.el	. . 3 |- ( ph -> B e. ( A ⊔ 1o ) )
24		eldju 6749	. . 3 |- ( B e. ( A ⊔ 1o ) <-> ( E. x e. A B = ( (inl |` A ) ` x ) \/ E. x e. 1o B = ( (inr |` 1o ) ` x ) ) )
25	23, 24	sylib 120	. 2 |- ( ph -> ( E. x e. A B = ( (inl |` A ) ` x ) \/ E. x e. 1o B = ( (inr |` 1o ) ` x ) ) )
26	14, 22, 25	mpjaodan 747	1 |- ( ph -> ( ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) \/ B = ( (inr |` 1o ) ` (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 664   = wceq 1289   e. wcel 1438   E.wrex 2360   C_ wss 2999   (/)c0 3286   |` cres 4438   ` cfv 5010   1oc1o 6166   ⊔ cdju 6720  inlcinl 6727  inrcinr 6728

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-suc 4196  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-1st 5903  df-2nd 5904  df-1o 6173  df-dju 6721  df-inl 6729  df-inr 6730

This theorem is referenced by:  exmidfodomrlemrALT  6819