Theorem ctiunct 12597

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 12601 (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 12599, 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 12552) 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 7170 and ctssdc 7172.

(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 12552	. . . . 5 ⊢ (ω × ω) ≈ ω
2	1	ensymi 6836	. . . 4 ⊢ ω ≈ (ω × ω)
3		bren 6801	. . . 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 2193	. . . . . . . 8 ⊢ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} = {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}
8		eqid 2193	. . . . . . . 8 ⊢ (◡inl ∘ 𝐹) = (◡inl ∘ 𝐹)
9	6, 7, 8	ctssdccl 7170	. . . . . . 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 1011	. . . . . 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 1013	. . . . . 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 1012	. . . . . 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 2193	. . . . . . . 8 ⊢ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} = {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
18		eqid 2193	. . . . . . . 8 ⊢ (◡inl ∘ 𝐺) = (◡inl ∘ 𝐺)
19	16, 17, 18	ctssdccl 7170	. . . . . . 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 1011	. . . . . 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 1013	. . . . . 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 1012	. . . . . 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 2193	. . . . 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 12593	. . . 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 2193	. . . . . 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 1539	. . . . . . . . 9 ⊢ Ⅎ𝑥(1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}
31		nfcsb1v 3113	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
32	31	nfel2 2349	. . . . . . . . 9 ⊢ Ⅎ𝑥(2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
33	30, 32	nfan 1576	. . . . . . . 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 2336	. . . . . . . 8 ⊢ Ⅎ𝑥ω
35	33, 34	nfrabw 2675	. . . . . . 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 3113	. . . . . . . 8 ⊢ Ⅎ𝑥⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)
37		nfcv 2336	. . . . . . . 8 ⊢ Ⅎ𝑥(2nd ‘(𝑗‘𝑛))
38	36, 37	nffv 5564	. . . . . . 7 ⊢ Ⅎ𝑥(⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))
39	35, 38	nfmpt 4121	. . . . . 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 12596	. . . . 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 4625	. . . . . . . 8 ⊢ ω ∈ V
42	41	rabex 4173	. . . . . . 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 5784	. . . . . 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 5472	. . . . . 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 2855	. . . . 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 12594	. . . 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 3202	. . . . . 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 5473	. . . . . . 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 1836	. . . . . 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 2257	. . . . . . . 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 839	. . . . . . 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 2494	. . . . . 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 1323	. . . . 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 2855	. . . 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 1249	. . 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 7172	. . . 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 5472	. . . . 5 ⊢ (𝑘 = ℎ → (𝑘:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o) ↔ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o)))
59	58	cbvexv 1930	. . . 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 1910	1 ⊢ (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835   ∧ w3a 980   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  {crab 2476  ⦋csb 3080   ⊆ wss 3153  ∪ ciun 3912   class class class wbr 4029   ↦ cmpt 4090  ωcom 4622   × cxp 4657  ^◡ccnv 4658   “ cima 4662   ∘ ccom 4663  –onto→wfo 5252  –1-1-onto→wf1o 5253  ‘cfv 5254  1^st c1st 6191  2^nd c2nd 6192  1_oc1o 6462   ≈ cen 6792   ⊔ cdju 7096  inlcinl 7104

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-dju 7097  df-inl 7106  df-inr 7107  df-case 7143  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-dvds 11931

This theorem is referenced by:  ctiunctal  12598  unct  12599