Theorem ctiunct 12395

Description: A sequence of enumerations gives an enumeration of the union. We refer to "sequence of enumerations" rather than "countably many countable sets" because the hypothesis provides more than countability for each 𝐵(𝑥): it refers to 𝐵(𝑥) together with the 𝐺(𝑥) which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74.

For "countably many countable sets" the key hypothesis would be (𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑔𝑔:ω–onto→(𝐵 ⊔ 1_o). This is almost omiunct 12399 (which uses countable choice) although that is for a countably infinite collection not any countable collection.

Compare with the case of two sets instead of countably many, as seen at unct 12397, which says that the union of two countable sets is countable .

The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 12350) and using the first number to map to an element 𝑥 of 𝐴 and the second number to map to an element of B(x) . In this way we are able to map to every element of ∪ 𝑥 ∈ 𝐴𝐵. Although it would be possible to work directly with countability expressed as 𝐹:ω–onto→(𝐴 ⊔ 1_o), we instead use functions from subsets of the natural numbers via ctssdccl 7088 and ctssdc 7090.

(Contributed by Jim Kingdon, 31-Oct-2023.)

Hypotheses

Ref	Expression
ctiunct.a	⊢ (𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1o))
ctiunct.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:ω–onto→(𝐵 ⊔ 1o))

Assertion

Ref	Expression
ctiunct	⊢ (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))

Distinct variable groups:   𝐴,ℎ,𝑥   𝐵,ℎ   𝑥,𝐹   𝜑,𝑥

Allowed substitution hints:   𝜑(ℎ)   𝐵(𝑥)   𝐹(ℎ)   𝐺(𝑥,ℎ)

Proof of Theorem ctiunct

Dummy variables 𝑗 𝑘 𝑛 𝑢 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpomen 12350	. . . . 5 ⊢ (ω × ω) ≈ ω
2	1	ensymi 6760	. . . 4 ⊢ ω ≈ (ω × ω)
3		bren 6725	. . . 4 ⊢ (ω ≈ (ω × ω) ↔ ∃𝑗 𝑗:ω–1-1-onto→(ω × ω))
4	2, 3	mpbi 144	. . 3 ⊢ ∃𝑗 𝑗:ω–1-1-onto→(ω × ω)
5	4	a1i 9	. 2 ⊢ (𝜑 → ∃𝑗 𝑗:ω–1-1-onto→(ω × ω))
6		ctiunct.a	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1o))
7		eqid 2170	. . . . . . . 8 ⊢ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} = {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}
8		eqid 2170	. . . . . . . 8 ⊢ (◡inl ∘ 𝐹) = (◡inl ∘ 𝐹)
9	6, 7, 8	ctssdccl 7088	. . . . . . 7 ⊢ (𝜑 → ({𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ⊆ ω ∧ (◡inl ∘ 𝐹):{𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}))
10	9	simp1d 1004	. . . . . 6 ⊢ (𝜑 → {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ⊆ ω)
11	10	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ⊆ ω)
12	9	simp3d 1006	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)})
13	12	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)})
14	9	simp2d 1005	. . . . . 6 ⊢ (𝜑 → (◡inl ∘ 𝐹):{𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}–onto→𝐴)
15	14	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → (◡inl ∘ 𝐹):{𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}–onto→𝐴)
16		ctiunct.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:ω–onto→(𝐵 ⊔ 1o))
17		eqid 2170	. . . . . . . 8 ⊢ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} = {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
18		eqid 2170	. . . . . . . 8 ⊢ (◡inl ∘ 𝐺) = (◡inl ∘ 𝐺)
19	16, 17, 18	ctssdccl 7088	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ({𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} ⊆ ω ∧ (◡inl ∘ 𝐺):{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}–onto→𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}))
20	19	simp1d 1004	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} ⊆ ω)
21	20	adantlr 474	. . . . 5 ⊢ (((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥 ∈ 𝐴) → {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} ⊆ ω)
22	19	simp3d 1006	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})
23	22	adantlr 474	. . . . 5 ⊢ (((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})
24	19	simp2d 1005	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡inl ∘ 𝐺):{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}–onto→𝐵)
25	24	adantlr 474	. . . . 5 ⊢ (((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥 ∈ 𝐴) → (◡inl ∘ 𝐺):{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}–onto→𝐵)
26		simpr 109	. . . . 5 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → 𝑗:ω–1-1-onto→(ω × ω))
27		eqid 2170	. . . . 5 ⊢ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} = {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}
28	11, 13, 15, 21, 23, 25, 26, 27	ctiunctlemuom 12391	. . . 4 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ⊆ ω)
29		eqid 2170	. . . . . 6 ⊢ (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ↦ (⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))) = (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ↦ (⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛))))
30		nfv 1521	. . . . . . . . 9 ⊢ Ⅎ𝑥(1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}
31		nfcsb1v 3082	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
32	31	nfel2 2325	. . . . . . . . 9 ⊢ Ⅎ𝑥(2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
33	30, 32	nfan 1558	. . . . . . . 8 ⊢ Ⅎ𝑥((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})
34		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑥ω
35	33, 34	nfrabxy 2650	. . . . . . 7 ⊢ Ⅎ𝑥{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}
36		nfcsb1v 3082	. . . . . . . 8 ⊢ Ⅎ𝑥⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)
37		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑥(2nd ‘(𝑗‘𝑛))
38	36, 37	nffv 5506	. . . . . . 7 ⊢ Ⅎ𝑥(⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))
39	35, 38	nfmpt 4081	. . . . . 6 ⊢ Ⅎ𝑥(𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ↦ (⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛))))
40	11, 13, 15, 21, 23, 25, 26, 27, 29, 39, 35	ctiunctlemfo 12394	. . . . 5 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ↦ (⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))):{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵)
41		omex 4577	. . . . . . . 8 ⊢ ω ∈ V
42	41	rabex 4133	. . . . . . 7 ⊢ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ∈ V
43	42	mptex 5722	. . . . . 6 ⊢ (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ↦ (⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))) ∈ V
44		foeq1 5416	. . . . . 6 ⊢ (𝑘 = (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ↦ (⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))) → (𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ↦ (⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))):{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵))
45	43, 44	spcev 2825	. . . . 5 ⊢ ((𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ↦ (⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))):{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵)
46	40, 45	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵)
47	11, 13, 15, 21, 23, 25, 26, 27	ctiunctlemudc 12392	. . . 4 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})})
48		sseq1 3170	. . . . . 6 ⊢ (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} → (𝑢 ⊆ ω ↔ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ⊆ ω))
49		foeq2 5417	. . . . . . 7 ⊢ (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} → (𝑘:𝑢–onto→∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵))
50	49	exbidv 1818	. . . . . 6 ⊢ (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} → (∃𝑘 𝑘:𝑢–onto→∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵))
51		eleq2 2234	. . . . . . . 8 ⊢ (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} → (𝑛 ∈ 𝑢 ↔ 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}))
52	51	dcbid 833	. . . . . . 7 ⊢ (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} → (DECID 𝑛 ∈ 𝑢 ↔ DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}))
53	52	ralbidv 2470	. . . . . 6 ⊢ (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} → (∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑢 ↔ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}))
54	48, 50, 53	3anbi123d 1307	. . . . 5 ⊢ (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} → ((𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢–onto→∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑢) ↔ ({𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ⊆ ω ∧ ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})})))
55	42, 54	spcev 2825	. . . 4 ⊢ (({𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})} ⊆ ω ∧ ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}–onto→∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})}) → ∃𝑢(𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢–onto→∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑢))
56	28, 46, 47, 55	syl3anc 1233	. . 3 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → ∃𝑢(𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢–onto→∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑢))
57		ctssdc 7090	. . . 4 ⊢ (∃𝑢(𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢–onto→∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑢) ↔ ∃𝑘 𝑘:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
58		foeq1 5416	. . . . 5 ⊢ (𝑘 = ℎ → (𝑘:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o) ↔ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o)))
59	58	cbvexv 1911	. . . 4 ⊢ (∃𝑘 𝑘:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o) ↔ ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
60	57, 59	bitri 183	. . 3 ⊢ (∃𝑢(𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢–onto→∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑢) ↔ ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
61	56, 60	sylib 121	. 2 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
62	5, 61	exlimddv 1891	1 ⊢ (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))

Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 829   ∧ w3a 973   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  {crab 2452  ⦋csb 3049   ⊆ wss 3121  ∪ ciun 3873   class class class wbr 3989   ↦ cmpt 4050  ωcom 4574   × cxp 4609  ^◡ccnv 4610   “ cima 4614   ∘ ccom 4615  –onto→wfo 5196  –1-1-onto→wf1o 5197  ‘cfv 5198  1^st c1st 6117  2^nd c2nd 6118  1_oc1o 6388   ≈ cen 6716   ⊔ cdju 7014  inlcinl 7022

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-er 6513  df-en 6719  df-dju 7015  df-inl 7024  df-inr 7025  df-case 7061  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-dvds 11750

This theorem is referenced by:  ctiunctal  12396  unct  12397