Theorem exmidfodomrlemreseldju 7410

Description: Lemma for exmidfodomrlemrALT 7413. A variant of eldju 7266. (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 3227	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. 1o )
3		el1o 6604	. . . . . . . . . 10 |- ( x e. 1o <-> x = (/) )
4	2, 3	sylib 122	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x = (/) )
5	4	fveq2d 5643	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( (inl |` A ) ` x ) = ( (inl |` A ) ` (/) ) )
6	5	eqeq2d 2243	. . . . . . 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 2309	. . . . . . . 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 2650	. . . 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 740	. 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 5643	. . . . . . 7 |- ( ( ph /\ x e. 1o ) -> ( (inr |` 1o ) ` x ) = ( (inr |` 1o ) ` (/) ) )
18	17	eqeq2d 2243	. . . . . 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 2650	. . . 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 741	. 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 7266	. . 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 805	1 |- ( ph -> ( ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) \/ B = ( (inr |` 1o ) ` (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   E.wrex 2511   C_ wss 3200   (/)c0 3494   |` cres 4727   ` cfv 5326   1oc1o 6574   ⊔ cdju 7235  inlcinl 7243  inrcinr 7244

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  exmidfodomrlemrALT  7413