Theorem exmidfodomrlemreseldju 7202

Description: Lemma for exmidfodomrlemrALT 7205. A variant of eldju 7070. (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 3157	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. 1o )
3		el1o 6441	. . . . . . . . . 10 |- ( x e. 1o <-> x = (/) )
4	2, 3	sylib 122	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x = (/) )
5	4	fveq2d 5521	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( (inl |` A ) ` x ) = ( (inl |` A ) ` (/) ) )
6	5	eqeq2d 2189	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( B = ( (inl |` A ) ` x ) <-> B = ( (inl |` A ) ` (/) ) ) )
7		simpr 110	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. A )
8	4, 7	eqeltrrd 2255	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> (/) e. A )
9	8	biantrurd 305	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( B = ( (inl |` A ) ` (/) ) <-> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) ) )
10	6, 9	bitrd 188	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( B = ( (inl |` A ) ` x ) <-> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) ) )
11	10	biimpd 144	. . . . 5 |- ( ( ph /\ x e. A ) -> ( B = ( (inl |` A ) ` x ) -> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) ) )
12	11	rexlimdva 2594	. . . 4 |- ( ph -> ( E. x e. A B = ( (inl |` A ) ` x ) -> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) ) )
13	12	imp 124	. . 3 |- ( ( ph /\ E. x e. A B = ( (inl |` A ) ` x ) ) -> ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) )
14	13	orcd 733	. 2 |- ( ( ph /\ E. x e. A B = ( (inl |` A ) ` x ) ) -> ( ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) \/ B = ( (inr |` 1o ) ` (/) ) ) )
15		simpr 110	. . . . . . . . 9 |- ( ( ph /\ x e. 1o ) -> x e. 1o )
16	15, 3	sylib 122	. . . . . . . 8 |- ( ( ph /\ x e. 1o ) -> x = (/) )
17	16	fveq2d 5521	. . . . . . 7 |- ( ( ph /\ x e. 1o ) -> ( (inr |` 1o ) ` x ) = ( (inr |` 1o ) ` (/) ) )
18	17	eqeq2d 2189	. . . . . 6 |- ( ( ph /\ x e. 1o ) -> ( B = ( (inr |` 1o ) ` x ) <-> B = ( (inr |` 1o ) ` (/) ) ) )
19	18	biimpd 144	. . . . 5 |- ( ( ph /\ x e. 1o ) -> ( B = ( (inr |` 1o ) ` x ) -> B = ( (inr |` 1o ) ` (/) ) ) )
20	19	rexlimdva 2594	. . . 4 |- ( ph -> ( E. x e. 1o B = ( (inr |` 1o ) ` x ) -> B = ( (inr |` 1o ) ` (/) ) ) )
21	20	imp 124	. . 3 |- ( ( ph /\ E. x e. 1o B = ( (inr |` 1o ) ` x ) ) -> B = ( (inr |` 1o ) ` (/) ) )
22	21	olcd 734	. 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 7070	. . 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 122	. 2 |- ( ph -> ( E. x e. A B = ( (inl |` A ) ` x ) \/ E. x e. 1o B = ( (inr |` 1o ) ` x ) ) )
26	14, 22, 25	mpjaodan 798	1 |- ( ph -> ( ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) \/ B = ( (inr |` 1o ) ` (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 708   = wceq 1353   e. wcel 2148   E.wrex 2456   C_ wss 3131   (/)c0 3424   |` cres 4630   ` cfv 5218   1oc1o 6413   ⊔ cdju 7039  inlcinl 7047  inrcinr 7048

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

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  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-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  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-1st 6144  df-2nd 6145  df-1o 6420  df-dju 7040  df-inl 7049  df-inr 7050

This theorem is referenced by:  exmidfodomrlemrALT  7205