Theorem exmidfodomrlemeldju 7502

Description: Lemma for exmidfodomr 7507. A variant of djur 7360. (Contributed by Jim Kingdon, 2-Jul-2022.)

Hypotheses

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

Assertion

Ref	Expression
exmidfodomrlemeldju	|- ( ph -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )

Proof of Theorem exmidfodomrlemeldju

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidfodomrlemeldju.a	. . . . . . . . . 10 |- ( ph -> A C_ 1o )
2	1	sselda 3238	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. 1o )
3		el1o 6670	. . . . . . . . 9 |- ( x e. 1o <-> x = (/) )
4	2, 3	sylib 122	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> x = (/) )
5	4	fveq2d 5674	. . . . . . 7 |- ( ( ph /\ x e. A ) -> (inl ` x ) = (inl ` (/) ) )
6	5	eqeq2d 2244	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( B = (inl ` x ) <-> B = (inl ` (/) ) ) )
7	6	biimpd 144	. . . . 5 |- ( ( ph /\ x e. A ) -> ( B = (inl ` x ) -> B = (inl ` (/) ) ) )
8	7	rexlimdva 2660	. . . 4 |- ( ph -> ( E. x e. A B = (inl ` x ) -> B = (inl ` (/) ) ) )
9	8	imp 124	. . 3 |- ( ( ph /\ E. x e. A B = (inl ` x ) ) -> B = (inl ` (/) ) )
10	9	orcd 741	. 2 |- ( ( ph /\ E. x e. A B = (inl ` x ) ) -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
11		simpr 110	. . . . . . . . 9 |- ( ( ph /\ x e. 1o ) -> x e. 1o )
12	11, 3	sylib 122	. . . . . . . 8 |- ( ( ph /\ x e. 1o ) -> x = (/) )
13	12	fveq2d 5674	. . . . . . 7 |- ( ( ph /\ x e. 1o ) -> (inr ` x ) = (inr ` (/) ) )
14	13	eqeq2d 2244	. . . . . 6 |- ( ( ph /\ x e. 1o ) -> ( B = (inr ` x ) <-> B = (inr ` (/) ) ) )
15	14	biimpd 144	. . . . 5 |- ( ( ph /\ x e. 1o ) -> ( B = (inr ` x ) -> B = (inr ` (/) ) ) )
16	15	rexlimdva 2660	. . . 4 |- ( ph -> ( E. x e. 1o B = (inr ` x ) -> B = (inr ` (/) ) ) )
17	16	imp 124	. . 3 |- ( ( ph /\ E. x e. 1o B = (inr ` x ) ) -> B = (inr ` (/) ) )
18	17	olcd 742	. 2 |- ( ( ph /\ E. x e. 1o B = (inr ` x ) ) -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )
19		exmidfodomrlemeldju.el	. . 3 |- ( ph -> B e. ( A ⊔ 1o ) )
20		djur 7360	. . 3 |- ( B e. ( A ⊔ 1o ) <-> ( E. x e. A B = (inl ` x ) \/ E. x e. 1o B = (inr ` x ) ) )
21	19, 20	sylib 122	. 2 |- ( ph -> ( E. x e. A B = (inl ` x ) \/ E. x e. 1o B = (inr ` x ) ) )
22	10, 18, 21	mpjaodan 806	1 |- ( ph -> ( B = (inl ` (/) ) \/ B = (inr ` (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203   E.wrex 2521   C_ wss 3211   (/)c0 3508   ` cfv 5352   1oc1o 6640   ⊔ cdju 7328  inlcinl 7336  inrcinr 7337

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1st 6334  df-2nd 6335  df-1o 6647  df-dju 7329  df-inl 7338  df-inr 7339

This theorem is referenced by:  exmidfodomrlemr  7505