Theorem difinfsnlem 6984

Description: Lemma for difinfsn 6985. The case where we need to swap B and (inr ` (/) ) in building the mapping G. (Contributed by Jim Kingdon, 9-Aug-2023.)

Hypotheses

Ref	Expression
difinfsnlem.dc	|- ( ph -> A. x e. A A. y e. A DECID x = y )
difinfsnlem.b	|- ( ph -> B e. A )
difinfsnlem.f	|- ( ph -> F : ( om ⊔ 1o ) -1-1-> A )
difinfsnlem.fb	|- ( ph -> ( F ` (inr ` (/) ) ) =/= B )
difinfsnlem.g	|- G = ( n e. om |-> if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) )

Assertion

Ref	Expression
difinfsnlem	|- ( ph -> G : om -1-1-> ( A \ { B } ) )

Distinct variable groups:   A, n, x, y   B, n, x, y   n, F, x, y   ph, n

Allowed substitution hints:   ph( x, y)   G( x, y, n)

Proof of Theorem difinfsnlem

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difinfsnlem.f	. . . . . . . 8 |- ( ph -> F : ( om ⊔ 1o ) -1-1-> A )
2		f1f 5328	. . . . . . . 8 |- ( F : ( om ⊔ 1o ) -1-1-> A -> F : ( om ⊔ 1o ) --> A )
3	1, 2	syl 14	. . . . . . 7 |- ( ph -> F : ( om ⊔ 1o ) --> A )
4		0lt1o 6337	. . . . . . . 8 |- (/) e. 1o
5		djurcl 6937	. . . . . . . 8 |- ( (/) e. 1o -> (inr ` (/) ) e. ( om ⊔ 1o ) )
6	4, 5	mp1i 10	. . . . . . 7 |- ( ph -> (inr ` (/) ) e. ( om ⊔ 1o ) )
7	3, 6	ffvelrnd 5556	. . . . . 6 |- ( ph -> ( F ` (inr ` (/) ) ) e. A )
8		difinfsnlem.fb	. . . . . . 7 |- ( ph -> ( F ` (inr ` (/) ) ) =/= B )
9		elsni 3545	. . . . . . . 8 |- ( ( F ` (inr ` (/) ) ) e. { B } -> ( F ` (inr ` (/) ) ) = B )
10	9	necon3ai 2357	. . . . . . 7 |- ( ( F ` (inr ` (/) ) ) =/= B -> -. ( F ` (inr ` (/) ) ) e. { B } )
11	8, 10	syl 14	. . . . . 6 |- ( ph -> -. ( F ` (inr ` (/) ) ) e. { B } )
12	7, 11	eldifd 3081	. . . . 5 |- ( ph -> ( F ` (inr ` (/) ) ) e. ( A \ { B } ) )
13	12	ad2antrr 479	. . . 4 |- ( ( ( ph /\ n e. om ) /\ ( F ` (inl ` n ) ) = B ) -> ( F ` (inr ` (/) ) ) e. ( A \ { B } ) )
14	3	adantr 274	. . . . . . 7 |- ( ( ph /\ n e. om ) -> F : ( om ⊔ 1o ) --> A )
15		djulcl 6936	. . . . . . . 8 |- ( n e. om -> (inl ` n ) e. ( om ⊔ 1o ) )
16	15	adantl 275	. . . . . . 7 |- ( ( ph /\ n e. om ) -> (inl ` n ) e. ( om ⊔ 1o ) )
17	14, 16	ffvelrnd 5556	. . . . . 6 |- ( ( ph /\ n e. om ) -> ( F ` (inl ` n ) ) e. A )
18	17	adantr 274	. . . . 5 |- ( ( ( ph /\ n e. om ) /\ -. ( F ` (inl ` n ) ) = B ) -> ( F ` (inl ` n ) ) e. A )
19		elsni 3545	. . . . . . 7 |- ( ( F ` (inl ` n ) ) e. { B } -> ( F ` (inl ` n ) ) = B )
20	19	con3i 621	. . . . . 6 |- ( -. ( F ` (inl ` n ) ) = B -> -. ( F ` (inl ` n ) ) e. { B } )
21	20	adantl 275	. . . . 5 |- ( ( ( ph /\ n e. om ) /\ -. ( F ` (inl ` n ) ) = B ) -> -. ( F ` (inl ` n ) ) e. { B } )
22	18, 21	eldifd 3081	. . . 4 |- ( ( ( ph /\ n e. om ) /\ -. ( F ` (inl ` n ) ) = B ) -> ( F ` (inl ` n ) ) e. ( A \ { B } ) )
23		difinfsnlem.dc	. . . . . 6 |- ( ph -> A. x e. A A. y e. A DECID x = y )
24	23	adantr 274	. . . . 5 |- ( ( ph /\ n e. om ) -> A. x e. A A. y e. A DECID x = y )
25		difinfsnlem.b	. . . . . . 7 |- ( ph -> B e. A )
26	25	adantr 274	. . . . . 6 |- ( ( ph /\ n e. om ) -> B e. A )
27		eqeq12 2152	. . . . . . . 8 |- ( ( x = ( F ` (inl ` n ) ) /\ y = B ) -> ( x = y <-> ( F ` (inl ` n ) ) = B ) )
28	27	dcbid 823	. . . . . . 7 |- ( ( x = ( F ` (inl ` n ) ) /\ y = B ) -> (DECID x = y <-> DECID ( F ` (inl ` n ) ) = B ) )
29	28	rspc2gv 2801	. . . . . 6 |- ( ( ( F ` (inl ` n ) ) e. A /\ B e. A ) -> ( A. x e. A A. y e. A DECID x = y -> DECID ( F ` (inl ` n ) ) = B ) )
30	17, 26, 29	syl2anc 408	. . . . 5 |- ( ( ph /\ n e. om ) -> ( A. x e. A A. y e. A DECID x = y -> DECID ( F ` (inl ` n ) ) = B ) )
31	24, 30	mpd 13	. . . 4 |- ( ( ph /\ n e. om ) -> DECID ( F ` (inl ` n ) ) = B )
32	13, 22, 31	ifcldadc 3501	. . 3 |- ( ( ph /\ n e. om ) -> if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) e. ( A \ { B } ) )
33	32	ralrimiva 2505	. 2 |- ( ph -> A. n e. om if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) e. ( A \ { B } ) )
34		simplr 519	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( F ` (inl ` n ) ) = B )
35		simpr 109	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( F ` (inl ` m ) ) = B )
36	34, 35	eqtr4d 2175	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( F ` (inl ` n ) ) = ( F ` (inl ` m ) ) )
37	1	ad3antrrr 483	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> F : ( om ⊔ 1o ) -1-1-> A )
38	15	ad2antrl 481	. . . . . . . . . 10 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> (inl ` n ) e. ( om ⊔ 1o ) )
39	38	ad2antrr 479	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> (inl ` n ) e. ( om ⊔ 1o ) )
40		simprr 521	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> m e. om )
41	40	ad2antrr 479	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> m e. om )
42		djulcl 6936	. . . . . . . . . 10 |- ( m e. om -> (inl ` m ) e. ( om ⊔ 1o ) )
43	41, 42	syl 14	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> (inl ` m ) e. ( om ⊔ 1o ) )
44		f1veqaeq 5670	. . . . . . . . 9 |- ( ( F : ( om ⊔ 1o ) -1-1-> A /\ ( (inl ` n ) e. ( om ⊔ 1o ) /\ (inl ` m ) e. ( om ⊔ 1o ) ) ) -> ( ( F ` (inl ` n ) ) = ( F ` (inl ` m ) ) -> (inl ` n ) = (inl ` m ) ) )
45	37, 39, 43, 44	syl12anc 1214	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( ( F ` (inl ` n ) ) = ( F ` (inl ` m ) ) -> (inl ` n ) = (inl ` m ) ) )
46	36, 45	mpd 13	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> (inl ` n ) = (inl ` m ) )
47		inl11 6950	. . . . . . . 8 |- ( ( n e. om /\ m e. om ) -> ( (inl ` n ) = (inl ` m ) <-> n = m ) )
48	47	ad3antlr 484	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( (inl ` n ) = (inl ` m ) <-> n = m ) )
49	46, 48	mpbid 146	. . . . . 6 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> n = m )
50	49	a1d 22	. . . . 5 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> n = m ) )
51	40	ad2antrr 479	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> m e. om )
52		djune 6963	. . . . . . . . . . 11 |- ( ( m e. om /\ (/) e. 1o ) -> (inl ` m ) =/= (inr ` (/) ) )
53	52	necomd 2394	. . . . . . . . . 10 |- ( ( m e. om /\ (/) e. 1o ) -> (inr ` (/) ) =/= (inl ` m ) )
54	51, 4, 53	sylancl 409	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> (inr ` (/) ) =/= (inl ` m ) )
55	54	neneqd 2329	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> -. (inr ` (/) ) = (inl ` m ) )
56	1	ad3antrrr 483	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> F : ( om ⊔ 1o ) -1-1-> A )
57	4, 5	mp1i 10	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> (inr ` (/) ) e. ( om ⊔ 1o ) )
58	40, 42	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> (inl ` m ) e. ( om ⊔ 1o ) )
59	58	ad2antrr 479	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> (inl ` m ) e. ( om ⊔ 1o ) )
60		f1veqaeq 5670	. . . . . . . . 9 |- ( ( F : ( om ⊔ 1o ) -1-1-> A /\ ( (inr ` (/) ) e. ( om ⊔ 1o ) /\ (inl ` m ) e. ( om ⊔ 1o ) ) ) -> ( ( F ` (inr ` (/) ) ) = ( F ` (inl ` m ) ) -> (inr ` (/) ) = (inl ` m ) ) )
61	56, 57, 59, 60	syl12anc 1214	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> ( ( F ` (inr ` (/) ) ) = ( F ` (inl ` m ) ) -> (inr ` (/) ) = (inl ` m ) ) )
62	55, 61	mtod 652	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> -. ( F ` (inr ` (/) ) ) = ( F ` (inl ` m ) ) )
63		simplr 519	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> ( F ` (inl ` n ) ) = B )
64	63	iftrued 3481	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = ( F ` (inr ` (/) ) ) )
65		simpr 109	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> -. ( F ` (inl ` m ) ) = B )
66	65	iffalsed 3484	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) = ( F ` (inl ` m ) ) )
67	64, 66	eqeq12d 2154	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) <-> ( F ` (inr ` (/) ) ) = ( F ` (inl ` m ) ) ) )
68	62, 67	mtbird 662	. . . . . 6 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> -. if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) )
69	68	pm2.21d 608	. . . . 5 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> n = m ) )
70	23	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> A. x e. A A. y e. A DECID x = y )
71	3	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> F : ( om ⊔ 1o ) --> A )
72	71, 58	ffvelrnd 5556	. . . . . . . . 9 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> ( F ` (inl ` m ) ) e. A )
73	25	adantr 274	. . . . . . . . 9 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> B e. A )
74		eqeq12 2152	. . . . . . . . . . 11 |- ( ( x = ( F ` (inl ` m ) ) /\ y = B ) -> ( x = y <-> ( F ` (inl ` m ) ) = B ) )
75	74	dcbid 823	. . . . . . . . . 10 |- ( ( x = ( F ` (inl ` m ) ) /\ y = B ) -> (DECID x = y <-> DECID ( F ` (inl ` m ) ) = B ) )
76	75	rspc2gv 2801	. . . . . . . . 9 |- ( ( ( F ` (inl ` m ) ) e. A /\ B e. A ) -> ( A. x e. A A. y e. A DECID x = y -> DECID ( F ` (inl ` m ) ) = B ) )
77	72, 73, 76	syl2anc 408	. . . . . . . 8 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> ( A. x e. A A. y e. A DECID x = y -> DECID ( F ` (inl ` m ) ) = B ) )
78	70, 77	mpd 13	. . . . . . 7 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> DECID ( F ` (inl ` m ) ) = B )
79		exmiddc 821	. . . . . . 7 |- (DECID ( F ` (inl ` m ) ) = B -> ( ( F ` (inl ` m ) ) = B \/ -. ( F ` (inl ` m ) ) = B ) )
80	78, 79	syl 14	. . . . . 6 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> ( ( F ` (inl ` m ) ) = B \/ -. ( F ` (inl ` m ) ) = B ) )
81	80	adantr 274	. . . . 5 |- ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) -> ( ( F ` (inl ` m ) ) = B \/ -. ( F ` (inl ` m ) ) = B ) )
82	50, 69, 81	mpjaodan 787	. . . 4 |- ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ ( F ` (inl ` n ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> n = m ) )
83		simprl 520	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> n e. om )
84	83	ad2antrr 479	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> n e. om )
85		djune 6963	. . . . . . . . . 10 |- ( ( n e. om /\ (/) e. 1o ) -> (inl ` n ) =/= (inr ` (/) ) )
86	84, 4, 85	sylancl 409	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> (inl ` n ) =/= (inr ` (/) ) )
87	86	neneqd 2329	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> -. (inl ` n ) = (inr ` (/) ) )
88	1	ad3antrrr 483	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> F : ( om ⊔ 1o ) -1-1-> A )
89	38	ad2antrr 479	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> (inl ` n ) e. ( om ⊔ 1o ) )
90	4, 5	mp1i 10	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> (inr ` (/) ) e. ( om ⊔ 1o ) )
91		f1veqaeq 5670	. . . . . . . . 9 |- ( ( F : ( om ⊔ 1o ) -1-1-> A /\ ( (inl ` n ) e. ( om ⊔ 1o ) /\ (inr ` (/) ) e. ( om ⊔ 1o ) ) ) -> ( ( F ` (inl ` n ) ) = ( F ` (inr ` (/) ) ) -> (inl ` n ) = (inr ` (/) ) ) )
92	88, 89, 90, 91	syl12anc 1214	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( ( F ` (inl ` n ) ) = ( F ` (inr ` (/) ) ) -> (inl ` n ) = (inr ` (/) ) ) )
93	87, 92	mtod 652	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> -. ( F ` (inl ` n ) ) = ( F ` (inr ` (/) ) ) )
94		simplr 519	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> -. ( F ` (inl ` n ) ) = B )
95	94	iffalsed 3484	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = ( F ` (inl ` n ) ) )
96		simpr 109	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( F ` (inl ` m ) ) = B )
97	96	iftrued 3481	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) = ( F ` (inr ` (/) ) ) )
98	95, 97	eqeq12d 2154	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) <-> ( F ` (inl ` n ) ) = ( F ` (inr ` (/) ) ) ) )
99	93, 98	mtbird 662	. . . . . 6 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> -. if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) )
100	99	pm2.21d 608	. . . . 5 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ ( F ` (inl ` m ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> n = m ) )
101		simplr 519	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> -. ( F ` (inl ` n ) ) = B )
102	101	iffalsed 3484	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = ( F ` (inl ` n ) ) )
103		simpr 109	. . . . . . . . 9 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> -. ( F ` (inl ` m ) ) = B )
104	103	iffalsed 3484	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) = ( F ` (inl ` m ) ) )
105	102, 104	eqeq12d 2154	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) <-> ( F ` (inl ` n ) ) = ( F ` (inl ` m ) ) ) )
106	1	ad3antrrr 483	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> F : ( om ⊔ 1o ) -1-1-> A )
107	38	ad2antrr 479	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> (inl ` n ) e. ( om ⊔ 1o ) )
108	58	ad2antrr 479	. . . . . . . 8 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> (inl ` m ) e. ( om ⊔ 1o ) )
109	106, 107, 108, 44	syl12anc 1214	. . . . . . 7 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> ( ( F ` (inl ` n ) ) = ( F ` (inl ` m ) ) -> (inl ` n ) = (inl ` m ) ) )
110	105, 109	sylbid 149	. . . . . 6 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> (inl ` n ) = (inl ` m ) ) )
111	47	ad3antlr 484	. . . . . 6 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> ( (inl ` n ) = (inl ` m ) <-> n = m ) )
112	110, 111	sylibd 148	. . . . 5 |- ( ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) /\ -. ( F ` (inl ` m ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> n = m ) )
113	80	adantr 274	. . . . 5 |- ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) -> ( ( F ` (inl ` m ) ) = B \/ -. ( F ` (inl ` m ) ) = B ) )
114	100, 112, 113	mpjaodan 787	. . . 4 |- ( ( ( ph /\ ( n e. om /\ m e. om ) ) /\ -. ( F ` (inl ` n ) ) = B ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> n = m ) )
115		exmiddc 821	. . . . . 6 |- (DECID ( F ` (inl ` n ) ) = B -> ( ( F ` (inl ` n ) ) = B \/ -. ( F ` (inl ` n ) ) = B ) )
116	31, 115	syl 14	. . . . 5 |- ( ( ph /\ n e. om ) -> ( ( F ` (inl ` n ) ) = B \/ -. ( F ` (inl ` n ) ) = B ) )
117	116	adantrr 470	. . . 4 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> ( ( F ` (inl ` n ) ) = B \/ -. ( F ` (inl ` n ) ) = B ) )
118	82, 114, 117	mpjaodan 787	. . 3 |- ( ( ph /\ ( n e. om /\ m e. om ) ) -> ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> n = m ) )
119	118	ralrimivva 2514	. 2 |- ( ph -> A. n e. om A. m e. om ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> n = m ) )
120		difinfsnlem.g	. . 3 |- G = ( n e. om |-> if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) )
121		2fveq3 5426	. . . . 5 |- ( n = m -> ( F ` (inl ` n ) ) = ( F ` (inl ` m ) ) )
122	121	eqeq1d 2148	. . . 4 |- ( n = m -> ( ( F ` (inl ` n ) ) = B <-> ( F ` (inl ` m ) ) = B ) )
123	122, 121	ifbieq2d 3496	. . 3 |- ( n = m -> if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) )
124	120, 123	f1mpt 5672	. 2 |- ( G : om -1-1-> ( A \ { B } ) <-> ( A. n e. om if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) e. ( A \ { B } ) /\ A. n e. om A. m e. om ( if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) = if ( ( F ` (inl ` m ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` m ) ) ) -> n = m ) ) )
125	33, 119, 124	sylanbrc 413	1 |- ( ph -> G : om -1-1-> ( A \ { B } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   =/= wne 2308   A.wral 2416   \ cdif 3068   (/)c0 3363   ifcif 3474   {csn 3527   |-> cmpt 3989   omcom 4504   -->wf 5119   -1-1->wf1 5120   ` cfv 5123   1oc1o 6306   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fv 5131  df-1st 6038  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933

This theorem is referenced by:  difinfsn  6985