Theorem difinfsnlem 7100

Description: Lemma for difinfsn 7101. 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 5423	. . . . . . . 8 ⊢ (𝐹:(ω ⊔ 1o)–1-1→𝐴 → 𝐹:(ω ⊔ 1o)⟶𝐴)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐹:(ω ⊔ 1o)⟶𝐴)
4		0lt1o 6443	. . . . . . . 8 ⊢ ∅ ∈ 1o
5		djurcl 7053	. . . . . . . 8 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (ω ⊔ 1o))
6	4, 5	mp1i 10	. . . . . . 7 ⊢ (𝜑 → (inr‘∅) ∈ (ω ⊔ 1o))
7	3, 6	ffvelcdmd 5654	. . . . . 6 ⊢ (𝜑 → (𝐹‘(inr‘∅)) ∈ 𝐴)
8		difinfsnlem.fb	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(inr‘∅)) ≠ 𝐵)
9		elsni 3612	. . . . . . . 8 ⊢ ((𝐹‘(inr‘∅)) ∈ {𝐵} → (𝐹‘(inr‘∅)) = 𝐵)
10	9	necon3ai 2396	. . . . . . 7 ⊢ ((𝐹‘(inr‘∅)) ≠ 𝐵 → ¬ (𝐹‘(inr‘∅)) ∈ {𝐵})
11	8, 10	syl 14	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘(inr‘∅)) ∈ {𝐵})
12	7, 11	eldifd 3141	. . . . 5 ⊢ (𝜑 → (𝐹‘(inr‘∅)) ∈ (𝐴 ∖ {𝐵}))
13	12	ad2antrr 488	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) → (𝐹‘(inr‘∅)) ∈ (𝐴 ∖ {𝐵}))
14	3	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → 𝐹:(ω ⊔ 1o)⟶𝐴)
15		djulcl 7052	. . . . . . . 8 ⊢ (𝑛 ∈ ω → (inl‘𝑛) ∈ (ω ⊔ 1o))
16	15	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (inl‘𝑛) ∈ (ω ⊔ 1o))
17	14, 16	ffvelcdmd 5654	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝐹‘(inl‘𝑛)) ∈ 𝐴)
18	17	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) → (𝐹‘(inl‘𝑛)) ∈ 𝐴)
19		elsni 3612	. . . . . . 7 ⊢ ((𝐹‘(inl‘𝑛)) ∈ {𝐵} → (𝐹‘(inl‘𝑛)) = 𝐵)
20	19	con3i 632	. . . . . 6 ⊢ (¬ (𝐹‘(inl‘𝑛)) = 𝐵 → ¬ (𝐹‘(inl‘𝑛)) ∈ {𝐵})
21	20	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) → ¬ (𝐹‘(inl‘𝑛)) ∈ {𝐵})
22	18, 21	eldifd 3141	. . . 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 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
25		difinfsnlem.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
26	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → 𝐵 ∈ 𝐴)
27		eqeq12 2190	. . . . . . . 8 ⊢ ((𝑥 = (𝐹‘(inl‘𝑛)) ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ (𝐹‘(inl‘𝑛)) = 𝐵))
28	27	dcbid 838	. . . . . . 7 ⊢ ((𝑥 = (𝐹‘(inl‘𝑛)) ∧ 𝑦 = 𝐵) → (DECID 𝑥 = 𝑦 ↔ DECID (𝐹‘(inl‘𝑛)) = 𝐵))
29	28	rspc2gv 2855	. . . . . 6 ⊢ (((𝐹‘(inl‘𝑛)) ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID (𝐹‘(inl‘𝑛)) = 𝐵))
30	17, 26, 29	syl2anc 411	. . . . 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 3565	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))) ∈ (𝐴 ∖ {𝐵}))
33	32	ralrimiva 2550	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ω if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))) ∈ (𝐴 ∖ {𝐵}))
34		simplr 528	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑛)) = 𝐵)
35		simpr 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑚)) = 𝐵)
36	34, 35	eqtr4d 2213	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑛)) = (𝐹‘(inl‘𝑚)))
37	1	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝐹:(ω ⊔ 1o)–1-1→𝐴)
38	15	ad2antrl 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → (inl‘𝑛) ∈ (ω ⊔ 1o))
39	38	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑛) ∈ (ω ⊔ 1o))
40		simprr 531	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → 𝑚 ∈ ω)
41	40	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝑚 ∈ ω)
42		djulcl 7052	. . . . . . . . . 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 5772	. . . . . . . . 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 1236	. . . . . . . 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 7066	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ 𝑚 ∈ ω) → ((inl‘𝑛) = (inl‘𝑚) ↔ 𝑛 = 𝑚))
48	47	ad3antlr 493	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ((inl‘𝑛) = (inl‘𝑚) ↔ 𝑛 = 𝑚))
49	46, 48	mpbid 147	. . . . . 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 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝑚 ∈ ω)
52		djune 7079	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ω ∧ ∅ ∈ 1o) → (inl‘𝑚) ≠ (inr‘∅))
53	52	necomd 2433	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ω ∧ ∅ ∈ 1o) → (inr‘∅) ≠ (inl‘𝑚))
54	51, 4, 53	sylancl 413	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (inr‘∅) ≠ (inl‘𝑚))
55	54	neneqd 2368	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (inr‘∅) = (inl‘𝑚))
56	1	ad3antrrr 492	. . . . . . . . 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 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑚) ∈ (ω ⊔ 1o))
60		f1veqaeq 5772	. . . . . . . . 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 1236	. . . . . . . 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 663	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inr‘∅)) = (𝐹‘(inl‘𝑚)))
63		simplr 528	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑛)) = 𝐵)
64	63	iftrued 3543	. . . . . . . 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 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑚)) = 𝐵)
66	65	iffalsed 3546	. . . . . . . 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 2192	. . . . . . 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 673	. . . . . 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 619	. . . . 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 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
71	3	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → 𝐹:(ω ⊔ 1o)⟶𝐴)
72	71, 58	ffvelcdmd 5654	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → (𝐹‘(inl‘𝑚)) ∈ 𝐴)
73	25	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → 𝐵 ∈ 𝐴)
74		eqeq12 2190	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝐹‘(inl‘𝑚)) ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ (𝐹‘(inl‘𝑚)) = 𝐵))
75	74	dcbid 838	. . . . . . . . . 10 ⊢ ((𝑥 = (𝐹‘(inl‘𝑚)) ∧ 𝑦 = 𝐵) → (DECID 𝑥 = 𝑦 ↔ DECID (𝐹‘(inl‘𝑚)) = 𝐵))
76	75	rspc2gv 2855	. . . . . . . . 9 ⊢ (((𝐹‘(inl‘𝑚)) ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → DECID (𝐹‘(inl‘𝑚)) = 𝐵))
77	72, 73, 76	syl2anc 411	. . . . . . . 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 836	. . . . . . 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 276	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ (𝐹‘(inl‘𝑛)) = 𝐵) → ((𝐹‘(inl‘𝑚)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑚)) = 𝐵))
82	50, 69, 81	mpjaodan 798	. . . 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 529	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → 𝑛 ∈ ω)
84	83	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝑛 ∈ ω)
85		djune 7079	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ ∅ ∈ 1o) → (inl‘𝑛) ≠ (inr‘∅))
86	84, 4, 85	sylancl 413	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑛) ≠ (inr‘∅))
87	86	neneqd 2368	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (inl‘𝑛) = (inr‘∅))
88	1	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝐹:(ω ⊔ 1o)–1-1→𝐴)
89	38	ad2antrr 488	. . . . . . . . 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 5772	. . . . . . . . 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 1236	. . . . . . . 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 663	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑛)) = (𝐹‘(inr‘∅)))
94		simplr 528	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑛)) = 𝐵)
95	94	iffalsed 3546	. . . . . . . 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 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ (𝐹‘(inl‘𝑚)) = 𝐵) → (𝐹‘(inl‘𝑚)) = 𝐵)
97	96	iftrued 3543	. . . . . . . 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 2192	. . . . . . 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 673	. . . . . 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 619	. . . . 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 528	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑛)) = 𝐵)
102	101	iffalsed 3546	. . . . . . . 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 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ¬ (𝐹‘(inl‘𝑚)) = 𝐵)
104	103	iffalsed 3546	. . . . . . . 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 2192	. . . . . . 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 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → 𝐹:(ω ⊔ 1o)–1-1→𝐴)
107	38	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑛) ∈ (ω ⊔ 1o))
108	58	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → (inl‘𝑚) ∈ (ω ⊔ 1o))
109	106, 107, 108, 44	syl12anc 1236	. . . . . . 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 150	. . . . . 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 493	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) ∧ ¬ (𝐹‘(inl‘𝑚)) = 𝐵) → ((inl‘𝑛) = (inl‘𝑚) ↔ 𝑛 = 𝑚))
112	110, 111	sylibd 149	. . . . 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 276	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) ∧ ¬ (𝐹‘(inl‘𝑛)) = 𝐵) → ((𝐹‘(inl‘𝑚)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑚)) = 𝐵))
114	100, 112, 113	mpjaodan 798	. . . 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 836	. . . . . 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 479	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝑚 ∈ ω)) → ((𝐹‘(inl‘𝑛)) = 𝐵 ∨ ¬ (𝐹‘(inl‘𝑛)) = 𝐵))
118	82, 114, 117	mpjaodan 798	. . 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 2559	. 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 5522	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐹‘(inl‘𝑛)) = (𝐹‘(inl‘𝑚)))
122	121	eqeq1d 2186	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝐹‘(inl‘𝑛)) = 𝐵 ↔ (𝐹‘(inl‘𝑚)) = 𝐵))
123	122, 121	ifbieq2d 3560	. . 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 5774	. 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 417	1 ⊢ (𝜑 → 𝐺:ω–1-1→(𝐴 ∖ {𝐵}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455   ∖ cdif 3128  ∅c0 3424  ifcif 3536  {csn 3594   ↦ cmpt 4066  ωcom 4591  ⟶wf 5214  –1-1→wf1 5215  ‘cfv 5218  1_oc1o 6412   ⊔ cdju 7038  inlcinl 7046  inrcinr 7047

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fv 5226  df-1st 6143  df-1o 6419  df-dju 7039  df-inl 7048  df-inr 7049

This theorem is referenced by:  difinfsn  7101