ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ctiunct GIF version

Theorem ctiunct 12440
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 (𝜑𝑥𝐴) → ∃𝑔𝑔:ω–onto→(𝐵 ⊔ 1o). This is almost omiunct 12444 (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 12442, 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 12395) 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 𝑥𝐴𝐵. Although it would be possible to work directly with countability expressed as 𝐹:ω–onto→(𝐴 ⊔ 1o), we instead use functions from subsets of the natural numbers via ctssdccl 7109 and ctssdc 7111.

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

Hypotheses
Ref Expression
ctiunct.a (𝜑𝐹:ω–onto→(𝐴 ⊔ 1o))
ctiunct.b ((𝜑𝑥𝐴) → 𝐺:ω–onto→(𝐵 ⊔ 1o))
Assertion
Ref Expression
ctiunct (𝜑 → ∃ :ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o))
Distinct variable groups:   𝐴,,𝑥   𝐵,   𝑥,𝐹   𝜑,𝑥
Allowed substitution hints:   𝜑()   𝐵(𝑥)   𝐹()   𝐺(𝑥,)

Proof of Theorem ctiunct
Dummy variables 𝑗 𝑘 𝑛 𝑢 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpomen 12395 . . . . 5 (ω × ω) ≈ ω
21ensymi 6781 . . . 4 ω ≈ (ω × ω)
3 bren 6746 . . . 4 (ω ≈ (ω × ω) ↔ ∃𝑗 𝑗:ω–1-1-onto→(ω × ω))
42, 3mpbi 145 . . 3 𝑗 𝑗:ω–1-1-onto→(ω × ω)
54a1i 9 . 2 (𝜑 → ∃𝑗 𝑗:ω–1-1-onto→(ω × ω))
6 ctiunct.a . . . . . . . 8 (𝜑𝐹:ω–onto→(𝐴 ⊔ 1o))
7 eqid 2177 . . . . . . . 8 {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} = {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)}
8 eqid 2177 . . . . . . . 8 (inl ∘ 𝐹) = (inl ∘ 𝐹)
96, 7, 8ctssdccl 7109 . . . . . . 7 (𝜑 → ({𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ⊆ ω ∧ (inl ∘ 𝐹):{𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)}–onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)}))
109simp1d 1009 . . . . . 6 (𝜑 → {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ⊆ ω)
1110adantr 276 . . . . 5 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ⊆ ω)
129simp3d 1011 . . . . . 6 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)})
1312adantr 276 . . . . 5 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)})
149simp2d 1010 . . . . . 6 (𝜑 → (inl ∘ 𝐹):{𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)}–onto𝐴)
1514adantr 276 . . . . 5 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → (inl ∘ 𝐹):{𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)}–onto𝐴)
16 ctiunct.b . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐺:ω–onto→(𝐵 ⊔ 1o))
17 eqid 2177 . . . . . . . 8 {𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)} = {𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)}
18 eqid 2177 . . . . . . . 8 (inl ∘ 𝐺) = (inl ∘ 𝐺)
1916, 17, 18ctssdccl 7109 . . . . . . 7 ((𝜑𝑥𝐴) → ({𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)} ⊆ ω ∧ (inl ∘ 𝐺):{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)}–onto𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)}))
2019simp1d 1009 . . . . . 6 ((𝜑𝑥𝐴) → {𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)} ⊆ ω)
2120adantlr 477 . . . . 5 (((𝜑𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥𝐴) → {𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)} ⊆ ω)
2219simp3d 1011 . . . . . 6 ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})
2322adantlr 477 . . . . 5 (((𝜑𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})
2419simp2d 1010 . . . . . 6 ((𝜑𝑥𝐴) → (inl ∘ 𝐺):{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)}–onto𝐵)
2524adantlr 477 . . . . 5 (((𝜑𝑗:ω–1-1-onto→(ω × ω)) ∧ 𝑥𝐴) → (inl ∘ 𝐺):{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)}–onto𝐵)
26 simpr 110 . . . . 5 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → 𝑗:ω–1-1-onto→(ω × ω))
27 eqid 2177 . . . . 5 {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} = {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}
2811, 13, 15, 21, 23, 25, 26, 27ctiunctlemuom 12436 . . . 4 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ⊆ ω)
29 eqid 2177 . . . . . 6 (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ↦ (((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)‘(2nd ‘(𝑗𝑛)))) = (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ↦ (((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)‘(2nd ‘(𝑗𝑛))))
30 nfv 1528 . . . . . . . . 9 𝑥(1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)}
31 nfcsb1v 3090 . . . . . . . . . 10 𝑥((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)}
3231nfel2 2332 . . . . . . . . 9 𝑥(2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)}
3330, 32nfan 1565 . . . . . . . 8 𝑥((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})
34 nfcv 2319 . . . . . . . 8 𝑥ω
3533, 34nfrabxy 2657 . . . . . . 7 𝑥{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}
36 nfcsb1v 3090 . . . . . . . 8 𝑥((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)
37 nfcv 2319 . . . . . . . 8 𝑥(2nd ‘(𝑗𝑛))
3836, 37nffv 5525 . . . . . . 7 𝑥(((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)‘(2nd ‘(𝑗𝑛)))
3935, 38nfmpt 4095 . . . . . 6 𝑥(𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ↦ (((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)‘(2nd ‘(𝑗𝑛))))
4011, 13, 15, 21, 23, 25, 26, 27, 29, 39, 35ctiunctlemfo 12439 . . . . 5 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ↦ (((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)‘(2nd ‘(𝑗𝑛)))):{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵)
41 omex 4592 . . . . . . . 8 ω ∈ V
4241rabex 4147 . . . . . . 7 {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ∈ V
4342mptex 5742 . . . . . 6 (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ↦ (((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)‘(2nd ‘(𝑗𝑛)))) ∈ V
44 foeq1 5434 . . . . . 6 (𝑘 = (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ↦ (((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)‘(2nd ‘(𝑗𝑛)))) → (𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵 ↔ (𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ↦ (((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)‘(2nd ‘(𝑗𝑛)))):{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵))
4543, 44spcev 2832 . . . . 5 ((𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ↦ (((inl ∘ 𝐹)‘(1st ‘(𝑗𝑛))) / 𝑥(inl ∘ 𝐺)‘(2nd ‘(𝑗𝑛)))):{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵 → ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵)
4640, 45syl 14 . . . 4 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵)
4711, 13, 15, 21, 23, 25, 26, 27ctiunctlemudc 12437 . . . 4 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})})
48 sseq1 3178 . . . . . 6 (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} → (𝑢 ⊆ ω ↔ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ⊆ ω))
49 foeq2 5435 . . . . . . 7 (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} → (𝑘:𝑢onto 𝑥𝐴 𝐵𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵))
5049exbidv 1825 . . . . . 6 (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} → (∃𝑘 𝑘:𝑢onto 𝑥𝐴 𝐵 ↔ ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵))
51 eleq2 2241 . . . . . . . 8 (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} → (𝑛𝑢𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}))
5251dcbid 838 . . . . . . 7 (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} → (DECID 𝑛𝑢DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}))
5352ralbidv 2477 . . . . . 6 (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} → (∀𝑛 ∈ ω DECID 𝑛𝑢 ↔ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}))
5448, 50, 533anbi123d 1312 . . . . 5 (𝑢 = {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} → ((𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢onto 𝑥𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑢) ↔ ({𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ⊆ ω ∧ ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})})))
5542, 54spcev 2832 . . . 4 (({𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})} ⊆ ω ∧ ∃𝑘 𝑘:{𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}–onto 𝑥𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑧 ∈ ω ∣ ((1st ‘(𝑗𝑧)) ∈ {𝑤 ∈ ω ∣ (𝐹𝑤) ∈ (inl “ 𝐴)} ∧ (2nd ‘(𝑗𝑧)) ∈ ((inl ∘ 𝐹)‘(1st ‘(𝑗𝑧))) / 𝑥{𝑤 ∈ ω ∣ (𝐺𝑤) ∈ (inl “ 𝐵)})}) → ∃𝑢(𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢onto 𝑥𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑢))
5628, 46, 47, 55syl3anc 1238 . . 3 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → ∃𝑢(𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢onto 𝑥𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑢))
57 ctssdc 7111 . . . 4 (∃𝑢(𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢onto 𝑥𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑢) ↔ ∃𝑘 𝑘:ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o))
58 foeq1 5434 . . . . 5 (𝑘 = → (𝑘:ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o) ↔ :ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o)))
5958cbvexv 1918 . . . 4 (∃𝑘 𝑘:ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o) ↔ ∃ :ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o))
6057, 59bitri 184 . . 3 (∃𝑢(𝑢 ⊆ ω ∧ ∃𝑘 𝑘:𝑢onto 𝑥𝐴 𝐵 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑢) ↔ ∃ :ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o))
6156, 60sylib 122 . 2 ((𝜑𝑗:ω–1-1-onto→(ω × ω)) → ∃ :ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o))
625, 61exlimddv 1898 1 (𝜑 → ∃ :ω–onto→( 𝑥𝐴 𝐵 ⊔ 1o))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 834  w3a 978   = wceq 1353  wex 1492  wcel 2148  wral 2455  {crab 2459  csb 3057  wss 3129   ciun 3886   class class class wbr 4003  cmpt 4064  ωcom 4589   × cxp 4624  ccnv 4625  cima 4629  ccom 4630  ontowfo 5214  1-1-ontowf1o 5215  cfv 5216  1st c1st 6138  2nd c2nd 6139  1oc1o 6409  cen 6737  cdju 7035  inlcinl 7043
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-1o 6416  df-er 6534  df-en 6740  df-dju 7036  df-inl 7045  df-inr 7046  df-case 7082  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-fz 10008  df-fl 10269  df-mod 10322  df-seqfrec 10445  df-exp 10519  df-dvds 11794
This theorem is referenced by:  ctiunctal  12441  unct  12442
  Copyright terms: Public domain W3C validator