Theorem exmidfodomrlemreseldju 7558

Description: Lemma for exmidfodomrlemrALT 7561. A variant of eldju 7414. (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 3248	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. 1o )
3		el1o 6714	. . . . . . . . . 10 |- ( x e. 1o <-> x = (/) )
4	2, 3	sylib 122	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x = (/) )
5	4	fveq2d 5702	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( (inl |` A ) ` x ) = ( (inl |` A ) ` (/) ) )
6	5	eqeq2d 2250	. . . . . . 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 2316	. . . . . . . 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 2668	. . . 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 745	. 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 5702	. . . . . . 7 |- ( ( ph /\ x e. 1o ) -> ( (inr |` 1o ) ` x ) = ( (inr |` 1o ) ` (/) ) )
18	17	eqeq2d 2250	. . . . . 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 2668	. . . 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 746	. 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 7414	. . 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 810	1 |- ( ph -> ( ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) \/ B = ( (inr |` 1o ) ` (/) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   (/)c0 3520   |` cres 4778   ` cfv 5380   1oc1o 6684   ⊔ cdju 7383  inlcinl 7391  inrcinr 7392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-1o 6691  df-dju 7384  df-inl 7393  df-inr 7394

This theorem is used by:  exmidfodomrlemrALT  7561