Theorem ctiunct 13212

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 13216 (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 13214, 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 13167) 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 7404 and ctssdc 7406.

(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 13167	. . . . 5 ⊢ (ω × ω) ≈ ω
2	1	ensymi 7024	. . . 4 ⊢ ω ≈ (ω × ω)
3		bren 6985	. . . 4 ⊢ (ω ≈ (ω × ω) ↔ ∃𝑗 𝑗:ω–1-1-onto→(ω × ω))
4	2, 3	mpbi 145	. . 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 2234	. . . . . . . 8 ⊢ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} = {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}
8		eqid 2234	. . . . . . . 8 ⊢ (◡inl ∘ 𝐹) = (◡inl ∘ 𝐹)
9	6, 7, 8	ctssdccl 7404	. . . . . . 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 1036	. . . . . 6 ⊢ (𝜑 → {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ⊆ ω)
11	10	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} ⊆ ω)
12	9	simp3d 1038	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)})
13	12	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)})
14	9	simp2d 1037	. . . . . 6 ⊢ (𝜑 → (◡inl ∘ 𝐹):{𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}–onto→𝐴)
15	14	adantr 276	. . . . 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 2234	. . . . . . . 8 ⊢ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} = {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
18		eqid 2234	. . . . . . . 8 ⊢ (◡inl ∘ 𝐺) = (◡inl ∘ 𝐺)
19	16, 17, 18	ctssdccl 7404	. . . . . . 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 1036	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} ⊆ ω)
21	20	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥 ∈ 𝐴) → {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} ⊆ ω)
22	19	simp3d 1038	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})
23	22	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})
24	19	simp2d 1037	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡inl ∘ 𝐺):{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}–onto→𝐵)
25	24	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥 ∈ 𝐴) → (◡inl ∘ 𝐺):{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}–onto→𝐵)
26		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → 𝑗:ω–1-1-onto→(ω × ω))
27		eqid 2234	. . . . 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 13208	. . . 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 2234	. . . . . 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 1577	. . . . . . . . 9 ⊢ Ⅎ𝑥(1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}
31		nfcsb1v 3173	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
32	31	nfel2 2399	. . . . . . . . 9 ⊢ Ⅎ𝑥(2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
33	30, 32	nfan 1614	. . . . . . . 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 2386	. . . . . . . 8 ⊢ Ⅎ𝑥ω
35	33, 34	nfrabw 2727	. . . . . . 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 3173	. . . . . . . 8 ⊢ Ⅎ𝑥⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)
37		nfcv 2386	. . . . . . . 8 ⊢ Ⅎ𝑥(2nd ‘(𝑗‘𝑛))
38	36, 37	nffv 5682	. . . . . . 7 ⊢ Ⅎ𝑥(⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))
39	35, 38	nfmpt 4204	. . . . . 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 13211	. . . . 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 4717	. . . . . . . 8 ⊢ ω ∈ V
42	41	rabex 4258	. . . . . . 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 5914	. . . . . 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 5588	. . . . . 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 2914	. . . . 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 13209	. . . 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 3263	. . . . . 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 5589	. . . . . . 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 1874	. . . . . 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 2298	. . . . . . . 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 846	. . . . . . 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 2544	. . . . . 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 1349	. . . . 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 2914	. . . 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 1274	. . 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 7406	. . . 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 5588	. . . . 5 ⊢ (𝑘 = ℎ → (𝑘:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o) ↔ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o)))
59	58	cbvexv 1970	. . . 4 ⊢ (∃𝑘 𝑘:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o) ↔ ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
60	57, 59	bitri 184	. . 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 122	. 2 ⊢ ((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))
62	5, 61	exlimddv 1950	1 ⊢ (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 842   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  {crab 2526  ⦋csb 3140   ⊆ wss 3213  ∪ ciun 3993   class class class wbr 4111   ↦ cmpt 4173  ωcom 4714   × cxp 4749  ^◡ccnv 4750   “ cima 4754   ∘ ccom 4755  –onto→wfo 5352  –1-1-onto→wf1o 5353  ‘cfv 5354  1^st c1st 6334  2^nd c2nd 6335  1_oc1o 6642   ≈ cen 6975   ⊔ cdju 7330  inlcinl 7338

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-en 6978  df-dju 7331  df-inl 7340  df-inr 7341  df-case 7377  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-fz 10349  df-fl 10637  df-mod 10692  df-seqfrec 10817  df-exp 10908  df-dvds 12482

This theorem is referenced by:  ctiunctal  13213  unct  13214