Theorem difinfsnlem 7064

Description: Lemma for difinfsn 7065. 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	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
difinfsnlem.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)
difinfsnlem.f	⊢ (𝜑 → 𝐹:(ω ⊔ 1o)–1-1→𝐴)
difinfsnlem.fb	⊢ (𝜑 → (𝐹‘(inr‘∅)) ≠ 𝐵)
difinfsnlem.g	⊢ 𝐺 = (𝑛 ∈ ω ↦ if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))))

Assertion

Ref	Expression
difinfsnlem	⊢ (𝜑 → 𝐺:ω–1-1→(𝐴 ∖ {𝐵}))

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

Allowed substitution hints:   𝜑(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 ⊢ (𝜑 → 𝐹:(ω ⊔ 1o)–1-1→𝐴)
2		f1f 5393	. . . . . . . 8 ⊢ (𝐹:(ω ⊔ 1o)–1-1→𝐴 → 𝐹:(ω ⊔ 1o)⟶𝐴)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐹:(ω ⊔ 1o)⟶𝐴)
4		0lt1o 6408	. . . . . . . 8 ⊢ ∅ ∈ 1o
5		djurcl 7017	. . . . . . . 8 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (ω ⊔ 1o))
6	4, 5	mp1i 10	. . . . . . 7 ⊢ (𝜑 → (inr‘∅) ∈ (ω ⊔ 1o))
7	3, 6	ffvelrnd 5621	. . . . . 6 ⊢ (𝜑 → (𝐹‘(inr‘∅)) ∈ 𝐴)
8		difinfsnlem.fb	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(inr‘∅)) ≠ 𝐵)
9		elsni 3594	. . . . . . . 8 ⊢ ((𝐹‘(inr‘∅)) ∈ {𝐵} → (𝐹‘(inr‘∅)) = 𝐵)
10	9	necon3ai 2385	. . . . . . 7 ⊢ ((𝐹‘(inr‘∅)) ≠ 𝐵 → ¬ (𝐹‘(inr‘∅)) ∈ {𝐵})
11	8, 10	syl 14	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘(inr‘∅)) ∈ {𝐵})
12	7, 11	eldifd 3126	. . . . 5 ⊢ (𝜑 → (𝐹‘(inr‘∅)) ∈ (𝐴 ∖ {𝐵}))
13	12	ad2antrr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) → (𝐹‘(inr‘∅)) ∈ (𝐴 ∖ {𝐵}))
14	3	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → 𝐹:(ω ⊔ 1o)⟶𝐴)
15		djulcl 7016	. . . . . . . 8 ⊢ (𝑛 ∈ ω → (inl‘𝑛) ∈ (ω ⊔ 1o))
16	15	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (inl‘𝑛) ∈ (ω ⊔ 1o))
17	14, 16	ffvelrnd 5621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝐹‘(inl‘𝑛)) ∈ 𝐴)
18	17	adantr 274	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) → (𝐹‘(inl‘𝑛)) ∈ 𝐴)
19		elsni 3594	. . . . . . 7 ⊢ ((𝐹‘(inl‘𝑛)) ∈ {𝐵} → (𝐹‘(inl‘𝑛)) = 𝐵)
20	19	con3i 622	. . . . . 6 ⊢ (¬ (𝐹‘(inl‘𝑛)) = 𝐵 → ¬ (𝐹‘(inl‘𝑛)) ∈ {𝐵})
21	20	adantl 275	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) → ¬ (𝐹‘(inl‘𝑛)) ∈ {𝐵})
22	18, 21	eldifd 3126	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) → (𝐹‘(inl‘𝑛)) ∈ (𝐴 ∖ {𝐵}))
23		difinfsnlem.dc	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
24	23	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
25		difinfsnlem.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
26	25	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → 𝐵 ∈ 𝐴)
27		eqeq12 2178	. . . . . . . 8 ⊢ ((𝑥 = (𝐹‘(inl‘𝑛)) ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ (𝐹‘(inl‘𝑛)) = 𝐵))
28	27	dcbid 828	. . . . . . 7 ⊢ ((𝑥 = (𝐹‘(inl‘𝑛)) ∧ 𝑦 = 𝐵) → (DECID 𝑥 = 𝑦 ↔ DECID (𝐹‘(inl‘𝑛)) = 𝐵))
29	28	rspc2gv 2842	. . . . . 6 ⊢ (((𝐹‘(inl‘𝑛)) ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID (𝐹‘(inl‘𝑛)) = 𝐵))
30	17, 26, 29	syl2anc 409	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID (𝐹‘(inl‘𝑛)) = 𝐵))
31	24, 30	mpd 13	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → DECID (𝐹‘(inl‘𝑛)) = 𝐵)
32	13, 22, 31	ifcldadc 3549	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))) ∈ (𝐴 ∖ {𝐵}))
33	32	ralrimiva 2539	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ω if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))) ∈ (𝐴 ∖ {𝐵}))
34		simplr 520	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑛)) = 𝐵)
35		simpr 109	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑚)) = 𝐵)
36	34, 35	eqtr4d 2201	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑛)) = (𝐹‘(inl‘𝑚)))
37	1	ad3antrrr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝐹:(ω ⊔ 1o)–1-1→𝐴)
38	15	ad2antrl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → (inl‘𝑛) ∈ (ω ⊔ 1o))
39	38	ad2antrr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑛) ∈ (ω ⊔ 1o))
40		simprr 522	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → 𝑚 ∈ ω)
41	40	ad2antrr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝑚 ∈ ω)
42		djulcl 7016	. . . . . . . . . 10 ⊢ (𝑚 ∈ ω → (inl‘𝑚) ∈ (ω ⊔ 1o))
43	41, 42	syl 14	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑚) ∈ (ω ⊔ 1o))
44		f1veqaeq 5737	. . . . . . . . 9 ⊢ ((𝐹:(ω ⊔ 1o)–1-1→𝐴 ∧ ((inl‘𝑛) ∈ (ω ⊔ 1o) ∧ (inl‘𝑚) ∈ (ω ⊔ 1o))) → ((𝐹‘(inl‘𝑛)) = (𝐹‘(inl‘𝑚)) → (inl‘𝑛) = (inl‘𝑚)))
45	37, 39, 43, 44	syl12anc 1226	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ((𝐹‘(inl‘𝑛)) = (𝐹‘(inl‘𝑚)) → (inl‘𝑛) = (inl‘𝑚)))
46	36, 45	mpd 13	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑛) = (inl‘𝑚))
47		inl11 7030	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ 𝑚 ∈ ω) → ((inl‘𝑛) = (inl‘𝑚) ↔ 𝑛 = 𝑚))
48	47	ad3antlr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ((inl‘𝑛) = (inl‘𝑚) ↔ 𝑛 = 𝑚))
49	46, 48	mpbid 146	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝑛 = 𝑚)
50	49	a1d 22	. . . . 5 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (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 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝑚 ∈ ω)
52		djune 7043	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ω ∧ ∅ ∈ 1o) → (inl‘𝑚) ≠ (inr‘∅))
53	52	necomd 2422	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ω ∧ ∅ ∈ 1o) → (inr‘∅) ≠ (inl‘𝑚))
54	51, 4, 53	sylancl 410	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (inr‘∅) ≠ (inl‘𝑚))
55	54	neneqd 2357	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (inr‘∅) = (inl‘𝑚))
56	1	ad3antrrr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝐹:(ω ⊔ 1o)–1-1→𝐴)
57	4, 5	mp1i 10	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (inr‘∅) ∈ (ω ⊔ 1o))
58	40, 42	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → (inl‘𝑚) ∈ (ω ⊔ 1o))
59	58	ad2antrr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑚) ∈ (ω ⊔ 1o))
60		f1veqaeq 5737	. . . . . . . . 9 ⊢ ((𝐹:(ω ⊔ 1o)–1-1→𝐴 ∧ ((inr‘∅) ∈ (ω ⊔ 1o) ∧ (inl‘𝑚) ∈ (ω ⊔ 1o))) → ((𝐹‘(inr‘∅)) = (𝐹‘(inl‘𝑚)) → (inr‘∅) = (inl‘𝑚)))
61	56, 57, 59, 60	syl12anc 1226	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ((𝐹‘(inr‘∅)) = (𝐹‘(inl‘𝑚)) → (inr‘∅) = (inl‘𝑚)))
62	55, 61	mtod 653	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inr‘∅)) = (𝐹‘(inl‘𝑚)))
63		simplr 520	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑛)) = 𝐵)
64	63	iftrued 3527	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))) = (𝐹‘(inr‘∅)))
65		simpr 109	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑚)) = 𝐵)
66	65	iffalsed 3530	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (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 2180	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (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 663	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (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 609	. . . . 5 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (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 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
71	3	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → 𝐹:(ω ⊔ 1o)⟶𝐴)
72	71, 58	ffvelrnd 5621	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → (𝐹‘(inl‘𝑚)) ∈ 𝐴)
73	25	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → 𝐵 ∈ 𝐴)
74		eqeq12 2178	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝐹‘(inl‘𝑚)) ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ (𝐹‘(inl‘𝑚)) = 𝐵))
75	74	dcbid 828	. . . . . . . . . 10 ⊢ ((𝑥 = (𝐹‘(inl‘𝑚)) ∧ 𝑦 = 𝐵) → (DECID 𝑥 = 𝑦 ↔ DECID (𝐹‘(inl‘𝑚)) = 𝐵))
76	75	rspc2gv 2842	. . . . . . . . 9 ⊢ (((𝐹‘(inl‘𝑚)) ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID (𝐹‘(inl‘𝑚)) = 𝐵))
77	72, 73, 76	syl2anc 409	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID (𝐹‘(inl‘𝑚)) = 𝐵))
78	70, 77	mpd 13	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → DECID (𝐹‘(inl‘𝑚)) = 𝐵)
79		exmiddc 826	. . . . . . 7 ⊢ (DECID (𝐹‘(inl‘𝑚)) = 𝐵 → ((𝐹‘(inl‘𝑚)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑚)) = 𝐵))
80	78, 79	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → ((𝐹‘(inl‘𝑚)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑚)) = 𝐵))
81	80	adantr 274	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) → ((𝐹‘(inl‘𝑚)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑚)) = 𝐵))
82	50, 69, 81	mpjaodan 788	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (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 521	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → 𝑛 ∈ ω)
84	83	ad2antrr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝑛 ∈ ω)
85		djune 7043	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ ∅ ∈ 1o) → (inl‘𝑛) ≠ (inr‘∅))
86	84, 4, 85	sylancl 410	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑛) ≠ (inr‘∅))
87	86	neneqd 2357	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (inl‘𝑛) = (inr‘∅))
88	1	ad3antrrr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝐹:(ω ⊔ 1o)–1-1→𝐴)
89	38	ad2antrr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑛) ∈ (ω ⊔ 1o))
90	4, 5	mp1i 10	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (inr‘∅) ∈ (ω ⊔ 1o))
91		f1veqaeq 5737	. . . . . . . . 9 ⊢ ((𝐹:(ω ⊔ 1o)–1-1→𝐴 ∧ ((inl‘𝑛) ∈ (ω ⊔ 1o) ∧ (inr‘∅) ∈ (ω ⊔ 1o))) → ((𝐹‘(inl‘𝑛)) = (𝐹‘(inr‘∅)) → (inl‘𝑛) = (inr‘∅)))
92	88, 89, 90, 91	syl12anc 1226	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ((𝐹‘(inl‘𝑛)) = (𝐹‘(inr‘∅)) → (inl‘𝑛) = (inr‘∅)))
93	87, 92	mtod 653	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑛)) = (𝐹‘(inr‘∅)))
94		simplr 520	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑛)) = 𝐵)
95	94	iffalsed 3530	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑚)) = 𝐵)
97	96	iftrued 3527	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → if((𝐹‘(inl‘𝑚)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑚))) = (𝐹‘(inr‘∅)))
98	95, 97	eqeq12d 2180	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 663	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 609	. . . . 5 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 520	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑛)) = 𝐵)
102	101	iffalsed 3530	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑚)) = 𝐵)
104	103	iffalsed 3530	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 2180	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝐹:(ω ⊔ 1o)–1-1→𝐴)
107	38	ad2antrr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑛) ∈ (ω ⊔ 1o))
108	58	ad2antrr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑚) ∈ (ω ⊔ 1o))
109	106, 107, 108, 44	syl12anc 1226	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ((𝐹‘(inl‘𝑛)) = (𝐹‘(inl‘𝑚)) → (inl‘𝑛) = (inl‘𝑚)))
110	105, 109	sylbid 149	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 485	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ((inl‘𝑛) = (inl‘𝑚) ↔ 𝑛 = 𝑚))
112	110, 111	sylibd 148	. . . . 5 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) → ((𝐹‘(inl‘𝑚)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑚)) = 𝐵))
114	100, 112, 113	mpjaodan 788	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (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 826	. . . . . 6 ⊢ (DECID (𝐹‘(inl‘𝑛)) = 𝐵 → ((𝐹‘(inl‘𝑛)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑛)) = 𝐵))
116	31, 115	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘(inl‘𝑛)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑛)) = 𝐵))
117	116	adantrr 471	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → ((𝐹‘(inl‘𝑛)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑛)) = 𝐵))
118	82, 114, 117	mpjaodan 788	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → (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 2548	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ω ∀𝑚 ∈ ω (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 ∈ ω ↦ if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))))
121		2fveq3 5491	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐹‘(inl‘𝑛)) = (𝐹‘(inl‘𝑚)))
122	121	eqeq1d 2174	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐹‘(inl‘𝑛)) = 𝐵 ↔ (𝐹‘(inl‘𝑚)) = 𝐵))
123	122, 121	ifbieq2d 3544	. . 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 5739	. 2 ⊢ (𝐺:ω–1-1→(𝐴 ∖ {𝐵}) ↔ (∀𝑛 ∈ ω if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))) ∈ (𝐴 ∖ {𝐵}) ∧ ∀𝑛 ∈ ω ∀𝑚 ∈ ω (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 414	1 ⊢ (𝜑 → 𝐺:ω–1-1→(𝐴 ∖ {𝐵}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 824   = wceq 1343   ∈ wcel 2136   ≠ wne 2336  ∀wral 2444   ∖ cdif 3113  ∅c0 3409  ifcif 3520  {csn 3576   ↦ cmpt 4043  ωcom 4567  ⟶wf 5184  –1-1→wf1 5185  ‘cfv 5188  1_oc1o 6377   ⊔ cdju 7002  inlcinl 7010  inrcinr 7011

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fv 5196  df-1st 6108  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013

This theorem is referenced by:  difinfsn  7065