Theorem exmidfodomrlemreseldju 7505

Description: Lemma for exmidfodomrlemrALT 7508. A variant of eldju 7361. (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 3240	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> x e. 1o )
3		el1o 6672	. . . . . . . . . 10 |- ( x e. 1o <-> x = (/) )
4	2, 3	sylib 122	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x = (/) )
5	4	fveq2d 5676	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( (inl |` A ) ` x ) = ( (inl |` A ) ` (/) ) )
6	5	eqeq2d 2246	. . . . . . 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 2312	. . . . . . . 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 2662	. . . 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 741	. 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 5676	. . . . . . 7 |- ( ( ph /\ x e. 1o ) -> ( (inr |` 1o ) ` x ) = ( (inr |` 1o ) ` (/) ) )
18	17	eqeq2d 2246	. . . . . 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 2662	. . . 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 742	. 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 7361	. . 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 806	1 |- ( ph -> ( ( (/) e. A /\ B = ( (inl |` A ) ` (/) ) ) \/ B = ( (inr |` 1o ) ` (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2205   E.wrex 2523   C_ wss 3213   (/)c0 3510   |` cres 4753   ` cfv 5354   1oc1o 6642   ⊔ cdju 7330  inlcinl 7338  inrcinr 7339

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-1st 6336  df-2nd 6337  df-1o 6649  df-dju 7331  df-inl 7340  df-inr 7341

This theorem is referenced by:  exmidfodomrlemrALT  7508