Theorem ctmlemr 6870

Description: Lemma for ctm 6871. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)

Assertion

Ref	Expression
ctmlemr	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))

Distinct variable groups:   𝐴,𝑓   𝑥,𝑓

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem ctmlemr

Dummy variables 𝑔 𝑛 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt1o 6242	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
2		djurcl 6824	. . . . . . . . . 10 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (𝐴 ⊔ 1o))
3	1, 2	ax-mp 7	. . . . . . . . 9 ⊢ (inr‘∅) ∈ (𝐴 ⊔ 1o)
4	3	a1i 9	. . . . . . . 8 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → (inr‘∅) ∈ (𝐴 ⊔ 1o))
5		simpllr 502	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → 𝑓:ω–onto→𝐴)
6		fof 5268	. . . . . . . . . . 11 ⊢ (𝑓:ω–onto→𝐴 → 𝑓:ω⟶𝐴)
7	5, 6	syl 14	. . . . . . . . . 10 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → 𝑓:ω⟶𝐴)
8		nnpredcl 4464	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ∪ 𝑛 ∈ ω)
9	8	ad2antlr 474	. . . . . . . . . 10 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → ∪ 𝑛 ∈ ω)
10	7, 9	ffvelrnd 5474	. . . . . . . . 9 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → (𝑓‘∪ 𝑛) ∈ 𝐴)
11		djulcl 6823	. . . . . . . . 9 ⊢ ((𝑓‘∪ 𝑛) ∈ 𝐴 → (inl‘(𝑓‘∪ 𝑛)) ∈ (𝐴 ⊔ 1o))
12	10, 11	syl 14	. . . . . . . 8 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → (inl‘(𝑓‘∪ 𝑛)) ∈ (𝐴 ⊔ 1o))
13		nndceq0 4459	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → DECID 𝑛 = ∅)
14	13	adantl 272	. . . . . . . 8 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) → DECID 𝑛 = ∅)
15	4, 12, 14	ifcldadc 3440	. . . . . . 7 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))) ∈ (𝐴 ⊔ 1o))
16	15	fmpttd 5492	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) → (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω⟶(𝐴 ⊔ 1o))
17		simpllr 502	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) → 𝑓:ω–onto→𝐴)
18		simprl 499	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) → 𝑤 ∈ 𝐴)
19		foelrn 5570	. . . . . . . . . . 11 ⊢ ((𝑓:ω–onto→𝐴 ∧ 𝑤 ∈ 𝐴) → ∃𝑢 ∈ ω 𝑤 = (𝑓‘𝑢))
20	17, 18, 19	syl2anc 404	. . . . . . . . . 10 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) → ∃𝑢 ∈ ω 𝑤 = (𝑓‘𝑢))
21		peano2 4438	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ω → suc 𝑢 ∈ ω)
22	21	ad2antrl 475	. . . . . . . . . . 11 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → suc 𝑢 ∈ ω)
23		simplrr 504	. . . . . . . . . . . . . 14 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 = (inl‘𝑤))
24		simprl 499	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑢 ∈ ω)
25		nnord 4454	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ω → Ord 𝑢)
26		ordtr 4229	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑢 → Tr 𝑢)
27	24, 25, 26	3syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → Tr 𝑢)
28		unisucg 4265	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ω → (Tr 𝑢 ↔ ∪ suc 𝑢 = 𝑢))
29	28	ad2antrl 475	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → (Tr 𝑢 ↔ ∪ suc 𝑢 = 𝑢))
30	27, 29	mpbid 146	. . . . . . . . . . . . . . . . 17 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → ∪ suc 𝑢 = 𝑢)
31	30	fveq2d 5344	. . . . . . . . . . . . . . . 16 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → (𝑓‘∪ suc 𝑢) = (𝑓‘𝑢))
32		simprr 500	. . . . . . . . . . . . . . . 16 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑤 = (𝑓‘𝑢))
33	31, 32	eqtr4d 2130	. . . . . . . . . . . . . . 15 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → (𝑓‘∪ suc 𝑢) = 𝑤)
34	33	fveq2d 5344	. . . . . . . . . . . . . 14 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → (inl‘(𝑓‘∪ suc 𝑢)) = (inl‘𝑤))
35	23, 34	eqtr4d 2130	. . . . . . . . . . . . 13 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 = (inl‘(𝑓‘∪ suc 𝑢)))
36		peano3 4439	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ ω → suc 𝑢 ≠ ∅)
37	36	neneqd 2283	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ω → ¬ suc 𝑢 = ∅)
38	37	ad2antrl 475	. . . . . . . . . . . . . 14 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → ¬ suc 𝑢 = ∅)
39	38	iffalsed 3423	. . . . . . . . . . . . 13 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))) = (inl‘(𝑓‘∪ suc 𝑢)))
40	35, 39	eqtr4d 2130	. . . . . . . . . . . 12 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))))
41		simpllr 502	. . . . . . . . . . . . . 14 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 ∈ (𝐴 ⊔ 1o))
42	40, 41	eqeltrrd 2172	. . . . . . . . . . . . 13 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))) ∈ (𝐴 ⊔ 1o))
43		eqeq1 2101	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = suc 𝑢 → (𝑛 = ∅ ↔ suc 𝑢 = ∅))
44		unieq 3684	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = suc 𝑢 → ∪ 𝑛 = ∪ suc 𝑢)
45	44	fveq2d 5344	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑢 → (𝑓‘∪ 𝑛) = (𝑓‘∪ suc 𝑢))
46	45	fveq2d 5344	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = suc 𝑢 → (inl‘(𝑓‘∪ 𝑛)) = (inl‘(𝑓‘∪ suc 𝑢)))
47	43, 46	ifbieq2d 3435	. . . . . . . . . . . . . 14 ⊢ (𝑛 = suc 𝑢 → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))) = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))))
48		eqid 2095	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))) = (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))
49	47, 48	fvmptg 5415	. . . . . . . . . . . . 13 ⊢ ((suc 𝑢 ∈ ω ∧ if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))) ∈ (𝐴 ⊔ 1o)) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘suc 𝑢) = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))))
50	22, 42, 49	syl2anc 404	. . . . . . . . . . . 12 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘suc 𝑢) = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))))
51	40, 50	eqtr4d 2130	. . . . . . . . . . 11 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘suc 𝑢))
52		fveq2 5340	. . . . . . . . . . . 12 ⊢ (𝑧 = suc 𝑢 → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧) = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘suc 𝑢))
53	52	rspceeqv 2753	. . . . . . . . . . 11 ⊢ ((suc 𝑢 ∈ ω ∧ 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘suc 𝑢)) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
54	22, 51, 53	syl2anc 404	. . . . . . . . . 10 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
55	20, 54	rexlimddv 2507	. . . . . . . . 9 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
56	55	rexlimdvaa 2503	. . . . . . . 8 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → (∃𝑤 ∈ 𝐴 𝑦 = (inl‘𝑤) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧)))
57		peano1 4437	. . . . . . . . . 10 ⊢ ∅ ∈ ω
58		simprr 500	. . . . . . . . . . . . 13 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑦 = (inr‘𝑤))
59		simprl 499	. . . . . . . . . . . . . . 15 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑤 ∈ 1o)
60		el1o 6239	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 1o ↔ 𝑤 = ∅)
61	59, 60	sylib 121	. . . . . . . . . . . . . 14 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑤 = ∅)
62	61	fveq2d 5344	. . . . . . . . . . . . 13 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → (inr‘𝑤) = (inr‘∅))
63	58, 62	eqtrd 2127	. . . . . . . . . . . 12 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑦 = (inr‘∅))
64		eqid 2095	. . . . . . . . . . . . 13 ⊢ ∅ = ∅
65	64	iftruei 3419	. . . . . . . . . . . 12 ⊢ if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))) = (inr‘∅)
66	63, 65	syl6eqr 2145	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑦 = if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))))
67	65, 3	eqeltri 2167	. . . . . . . . . . . . 13 ⊢ if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))) ∈ (𝐴 ⊔ 1o)
68	67	a1i 9	. . . . . . . . . . . 12 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))) ∈ (𝐴 ⊔ 1o))
69		eqeq1 2101	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ∅ → (𝑛 = ∅ ↔ ∅ = ∅))
70		unieq 3684	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ∅ → ∪ 𝑛 = ∪ ∅)
71	70	fveq2d 5344	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ∅ → (𝑓‘∪ 𝑛) = (𝑓‘∪ ∅))
72	71	fveq2d 5344	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ∅ → (inl‘(𝑓‘∪ 𝑛)) = (inl‘(𝑓‘∪ ∅)))
73	69, 72	ifbieq2d 3435	. . . . . . . . . . . . 13 ⊢ (𝑛 = ∅ → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))))
74	73, 48	fvmptg 5415	. . . . . . . . . . . 12 ⊢ ((∅ ∈ ω ∧ if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))) ∈ (𝐴 ⊔ 1o)) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))))
75	57, 68, 74	sylancr 406	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))))
76	66, 75	eqtr4d 2130	. . . . . . . . . 10 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅))
77		fveq2 5340	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧) = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅))
78	77	rspceeqv 2753	. . . . . . . . . 10 ⊢ ((∅ ∈ ω ∧ 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅)) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
79	57, 76, 78	sylancr 406	. . . . . . . . 9 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
80	79	rexlimdvaa 2503	. . . . . . . 8 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧)))
81		djur 6837	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐴 ⊔ 1o) → (∃𝑤 ∈ 𝐴 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
82	81	adantl 272	. . . . . . . 8 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → (∃𝑤 ∈ 𝐴 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
83	56, 80, 82	mpjaod 676	. . . . . . 7 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
84	83	ralrimiva 2458	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) → ∀𝑦 ∈ (𝐴 ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
85		dffo3 5485	. . . . . 6 ⊢ ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω–onto→(𝐴 ⊔ 1o) ↔ ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω⟶(𝐴 ⊔ 1o) ∧ ∀𝑦 ∈ (𝐴 ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧)))
86	16, 84, 85	sylanbrc 409	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) → (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω–onto→(𝐴 ⊔ 1o))
87		omex 4436	. . . . . . 7 ⊢ ω ∈ V
88	87	mptex 5562	. . . . . 6 ⊢ (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))) ∈ V
89		foeq1 5264	. . . . . 6 ⊢ (𝑔 = (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))) → (𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω–onto→(𝐴 ⊔ 1o)))
90	88, 89	spcev 2727	. . . . 5 ⊢ ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
91	86, 90	syl 14	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
92	91	ex 114	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑓:ω–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
93	92	exlimdv 1754	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
94		foeq1 5264	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
95	94	cbvexv 1850	. 2 ⊢ (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
96	93, 95	syl6ibr 161	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 667  DECID wdc 783   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371  ∅c0 3302  ifcif 3413  ∪ cuni 3675   ↦ cmpt 3921  Tr wtr 3958  Ord word 4213  suc csuc 4216  ωcom 4433  ⟶wf 5045  –onto→wfo 5047  ‘cfv 5049  1_oc1o 6212   ⊔ cdju 6810  inlcinl 6817  inrcinr 6818

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950  df-1o 6219  df-dju 6811  df-inl 6819  df-inr 6820

This theorem is referenced by:  ctm  6871