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Theorem ptcmplem4 22660
Description: Lemma for ptcmp 22663. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
ptcmp.1 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
ptcmp.2 𝑋 = X𝑛𝐴 (𝐹𝑛)
ptcmp.3 (𝜑𝐴𝑉)
ptcmp.4 (𝜑𝐹:𝐴⟶Comp)
ptcmp.5 (𝜑𝑋 ∈ (UFL ∩ dom card))
ptcmplem2.5 (𝜑𝑈 ⊆ ran 𝑆)
ptcmplem2.6 (𝜑𝑋 = 𝑈)
ptcmplem2.7 (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
ptcmplem3.8 𝐾 = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈}
Assertion
Ref Expression
ptcmplem4 ¬ 𝜑
Distinct variable groups:   𝑘,𝑛,𝑢,𝑤,𝑧,𝐴   𝑢,𝐾   𝑆,𝑘,𝑛,𝑢,𝑧   𝜑,𝑘,𝑛,𝑢   𝑈,𝑘,𝑢,𝑧   𝑘,𝑉,𝑛,𝑢,𝑤,𝑧   𝑘,𝐹,𝑛,𝑢,𝑤,𝑧   𝑘,𝑋,𝑛,𝑢,𝑤,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝑆(𝑤)   𝑈(𝑤,𝑛)   𝐾(𝑧,𝑤,𝑘,𝑛)

Proof of Theorem ptcmplem4
Dummy variables 𝑓 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcmp.1 . . 3 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
2 ptcmp.2 . . 3 𝑋 = X𝑛𝐴 (𝐹𝑛)
3 ptcmp.3 . . 3 (𝜑𝐴𝑉)
4 ptcmp.4 . . 3 (𝜑𝐹:𝐴⟶Comp)
5 ptcmp.5 . . 3 (𝜑𝑋 ∈ (UFL ∩ dom card))
6 ptcmplem2.5 . . 3 (𝜑𝑈 ⊆ ran 𝑆)
7 ptcmplem2.6 . . 3 (𝜑𝑋 = 𝑈)
8 ptcmplem2.7 . . 3 (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
9 ptcmplem3.8 . . 3 𝐾 = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈}
101, 2, 3, 4, 5, 6, 7, 8, 9ptcmplem3 22659 . 2 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
11 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓 Fn 𝐴)
12 eldifi 4054 . . . . . . . . . . . 12 ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → (𝑓𝑘) ∈ (𝐹𝑘))
1312ralimi 3128 . . . . . . . . . . 11 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
14 fveq2 6645 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
15 fveq2 6645 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
1615unieqd 4814 . . . . . . . . . . . . 13 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
1714, 16eleq12d 2884 . . . . . . . . . . . 12 (𝑛 = 𝑘 → ((𝑓𝑛) ∈ (𝐹𝑛) ↔ (𝑓𝑘) ∈ (𝐹𝑘)))
1817cbvralvw 3396 . . . . . . . . . . 11 (∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛) ↔ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
1913, 18sylibr 237 . . . . . . . . . 10 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
2019ad2antll 728 . . . . . . . . 9 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
21 vex 3444 . . . . . . . . . 10 𝑓 ∈ V
2221elixp 8451 . . . . . . . . 9 (𝑓X𝑛𝐴 (𝐹𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛)))
2311, 20, 22sylanbrc 586 . . . . . . . 8 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓X𝑛𝐴 (𝐹𝑛))
2423, 2eleqtrrdi 2901 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓𝑋)
257adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑋 = 𝑈)
2624, 25eleqtrd 2892 . . . . . 6 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓 𝑈)
27 eluni2 4804 . . . . . 6 (𝑓 𝑈 ↔ ∃𝑣𝑈 𝑓𝑣)
2826, 27sylib 221 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∃𝑣𝑈 𝑓𝑣)
29 simplrr 777 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓𝑣)
3029adantr 484 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑓𝑣)
31 simprr 772 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
3230, 31eleqtrd 2892 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
33 fveq1 6644 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑓 → (𝑤𝑘) = (𝑓𝑘))
3433eleq1d 2874 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑓 → ((𝑤𝑘) ∈ 𝑢 ↔ (𝑓𝑘) ∈ 𝑢))
35 eqid 2798 . . . . . . . . . . . . . . . . . 18 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
3635mptpreima 6059 . . . . . . . . . . . . . . . . 17 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢}
3734, 36elrab2 3631 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑓𝑋 ∧ (𝑓𝑘) ∈ 𝑢))
3837simprbi 500 . . . . . . . . . . . . . . 15 (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝑢)
3932, 38syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → (𝑓𝑘) ∈ 𝑢)
40 simprl 770 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢 ∈ (𝐹𝑘))
41 simplrl 776 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑣𝑈)
4241adantr 484 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑣𝑈)
4331, 42eqeltrrd 2891 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈)
44 rabid 3331 . . . . . . . . . . . . . . . 16 (𝑢 ∈ {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈} ↔ (𝑢 ∈ (𝐹𝑘) ∧ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈))
4540, 43, 44sylanbrc 586 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢 ∈ {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈})
4645, 9eleqtrrdi 2901 . . . . . . . . . . . . . 14 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢𝐾)
47 elunii 4805 . . . . . . . . . . . . . 14 (((𝑓𝑘) ∈ 𝑢𝑢𝐾) → (𝑓𝑘) ∈ 𝐾)
4839, 46, 47syl2anc 587 . . . . . . . . . . . . 13 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → (𝑓𝑘) ∈ 𝐾)
4948rexlimdvaa 3244 . . . . . . . . . . . 12 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾))
5049expr 460 . . . . . . . . . . 11 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ 𝑘𝐴) → ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5150ralimdva 3144 . . . . . . . . . 10 (((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) → (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5251ex 416 . . . . . . . . 9 ((𝜑𝑓 Fn 𝐴) → ((𝑣𝑈𝑓𝑣) → (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾))))
5352com23 86 . . . . . . . 8 ((𝜑𝑓 Fn 𝐴) → (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ((𝑣𝑈𝑓𝑣) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾))))
5453impr 458 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ((𝑣𝑈𝑓𝑣) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5554imp 410 . . . . . 6 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾))
566adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑈 ⊆ ran 𝑆)
5756sselda 3915 . . . . . . . . 9 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ 𝑣𝑈) → 𝑣 ∈ ran 𝑆)
5857adantrr 716 . . . . . . . 8 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → 𝑣 ∈ ran 𝑆)
591rnmpo 7263 . . . . . . . 8 ran 𝑆 = {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)}
6058, 59eleqtrdi 2900 . . . . . . 7 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → 𝑣 ∈ {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)})
61 abid 2780 . . . . . . 7 (𝑣 ∈ {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)} ↔ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
6260, 61sylib 221 . . . . . 6 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
63 rexim 3204 . . . . . 6 (∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾) → (∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾))
6455, 62, 63sylc 65 . . . . 5 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
6528, 64rexlimddv 3250 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
66 eldifn 4055 . . . . . . 7 ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ¬ (𝑓𝑘) ∈ 𝐾)
6766ralimi 3128 . . . . . 6 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾)
6867ad2antll 728 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾)
69 ralnex 3199 . . . . 5 (∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾 ↔ ¬ ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
7068, 69sylib 221 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ¬ ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
7165, 70pm2.65da 816 . . 3 (𝜑 → ¬ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
7271nexdv 1937 . 2 (𝜑 → ¬ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
7310, 72pm2.65i 197 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2111  {cab 2776  wral 3106  wrex 3107  {crab 3110  cdif 3878  cin 3880  wss 3881  𝒫 cpw 4497   cuni 4800  cmpt 5110  ccnv 5518  dom cdm 5519  ran crn 5520  cima 5522   Fn wfn 6319  wf 6320  cfv 6324  cmpo 7137  Xcixp 8444  Fincfn 8492  cardccrd 9348  Compccmp 21991  UFLcufl 22505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-fin 8496  df-wdom 9013  df-card 9352  df-acn 9355  df-cmp 21992
This theorem is referenced by:  ptcmplem5  22661
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