Theorem ctiunct 13390

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 13394 (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 13392, 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 13345) 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 7458 and ctssdc 7460.

(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 13345	. . . . 5 ⊢ (ω × ω) ≈ ω
2	1	ensymi 7074	. . . 4 ⊢ ω ≈ (ω × ω)
3		bren 7035	. . . 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 2238	. . . . . . . 8 ⊢ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)} = {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}
8		eqid 2238	. . . . . . . 8 ⊢ (◡inl ∘ 𝐹) = (◡inl ∘ 𝐹)
9	6, 7, 8	ctssdccl 7458	. . . . . . 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 1040	. . . . . 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 1042	. . . . . 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 1041	. . . . . 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 2238	. . . . . . . 8 ⊢ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} = {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
18		eqid 2238	. . . . . . . 8 ⊢ (◡inl ∘ 𝐺) = (◡inl ∘ 𝐺)
19	16, 17, 18	ctssdccl 7458	. . . . . . 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 1040	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} ⊆ ω)
21	20	adantlr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥 ∈ 𝐴) → {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)} ⊆ ω)
22	19	simp3d 1042	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})
23	22	adantlr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)})
24	19	simp2d 1041	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡inl ∘ 𝐺):{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}–onto→𝐵)
25	24	adantlr 481	. . . . 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 2238	. . . . 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 13386	. . . 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 2238	. . . . . 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 1581	. . . . . . . . 9 ⊢ Ⅎ𝑥(1st ‘(𝑗‘𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹‘𝑤) ∈ (inl “ 𝐴)}
31		nfcsb1v 3180	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
32	31	nfel2 2405	. . . . . . . . 9 ⊢ Ⅎ𝑥(2nd ‘(𝑗‘𝑧)) ∈ ⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑧))) / 𝑥⦌{𝑤 ∈ ω ∣ (𝐺‘𝑤) ∈ (inl “ 𝐵)}
33	30, 32	nfan 1618	. . . . . . . 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 2392	. . . . . . . 8 ⊢ Ⅎ𝑥ω
35	33, 34	nfrabw 2733	. . . . . . 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 3180	. . . . . . . 8 ⊢ Ⅎ𝑥⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)
37		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥(2nd ‘(𝑗‘𝑛))
38	36, 37	nffv 5708	. . . . . . 7 ⊢ Ⅎ𝑥(⦋((◡inl ∘ 𝐹)‘(1st ‘(𝑗‘𝑛))) / 𝑥⦌(◡inl ∘ 𝐺)‘(2nd ‘(𝑗‘𝑛)))
39	35, 38	nfmpt 4225	. . . . . 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 13389	. . . . 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 4742	. . . . . . . 8 ⊢ ω ∈ V
42	41	rabex 4282	. . . . . . 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 5946	. . . . . 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 5614	. . . . . 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 2920	. . . . 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 13387	. . . 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 3271	. . . . . 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 5615	. . . . . . 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 1878	. . . . . 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 2302	. . . . . . . 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 850	. . . . . . 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 2550	. . . . . 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 1353	. . . . 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 2920	. . . 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 1278	. . 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 7460	. . . 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 5614	. . . . 5 ⊢ (𝑘 = ℎ → (𝑘:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o) ↔ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o)))
59	58	cbvexv 1974	. . . 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 1954	1 ⊢ (𝜑 → ∃ℎ ℎ:ω–onto→(∪ 𝑥 ∈ 𝐴 𝐵 ⊔ 1o))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  DECID wdc 846   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  ⦋csb 3147   ⊆ wss 3220  ∪ ciun 4012   class class class wbr 4131   ↦ cmpt 4193  ωcom 4739   × cxp 4774  ^◡ccnv 4775   “ cima 4779   ∘ ccom 4780  –onto→wfo 5378  –1-1-onto→wf1o 5379  ‘cfv 5380  1^st c1st 6376  2^nd c2nd 6377  1_oc1o 6684   ≈ cen 7025   ⊔ cdju 7384  inlcinl 7392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7028  df-dju 7385  df-inl 7394  df-inr 7395  df-case 7431  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-fz 10430  df-fl 10723  df-mod 10782  df-seqfrec 10907  df-exp 10998  df-dvds 12581

This theorem is used by:  ctiunctal  13391  unct  13392