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Theorem ptcmplem4 21782
Description: Lemma for ptcmp 21785. (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 21781 . 2 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
11 simprl 793 . . . . . . . . 9 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓 Fn 𝐴)
12 eldifi 3715 . . . . . . . . . . . 12 ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → (𝑓𝑘) ∈ (𝐹𝑘))
1312ralimi 2947 . . . . . . . . . . 11 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
14 fveq2 6153 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
15 fveq2 6153 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
1615unieqd 4417 . . . . . . . . . . . . 13 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
1714, 16eleq12d 2692 . . . . . . . . . . . 12 (𝑛 = 𝑘 → ((𝑓𝑛) ∈ (𝐹𝑛) ↔ (𝑓𝑘) ∈ (𝐹𝑘)))
1817cbvralv 3162 . . . . . . . . . . 11 (∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛) ↔ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
1913, 18sylibr 224 . . . . . . . . . 10 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
2019ad2antll 764 . . . . . . . . 9 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
21 vex 3192 . . . . . . . . . 10 𝑓 ∈ V
2221elixp 7867 . . . . . . . . 9 (𝑓X𝑛𝐴 (𝐹𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛)))
2311, 20, 22sylanbrc 697 . . . . . . . 8 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓X𝑛𝐴 (𝐹𝑛))
2423, 2syl6eleqr 2709 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓𝑋)
257adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑋 = 𝑈)
2624, 25eleqtrd 2700 . . . . . 6 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓 𝑈)
27 eluni2 4411 . . . . . 6 (𝑓 𝑈 ↔ ∃𝑣𝑈 𝑓𝑣)
2826, 27sylib 208 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∃𝑣𝑈 𝑓𝑣)
29 simplrr 800 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓𝑣)
3029adantr 481 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑓𝑣)
31 simprr 795 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
3230, 31eleqtrd 2700 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
33 fveq1 6152 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑓 → (𝑤𝑘) = (𝑓𝑘))
3433eleq1d 2683 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑓 → ((𝑤𝑘) ∈ 𝑢 ↔ (𝑓𝑘) ∈ 𝑢))
35 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
3635mptpreima 5592 . . . . . . . . . . . . . . . . 17 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢}
3734, 36elrab2 3352 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑓𝑋 ∧ (𝑓𝑘) ∈ 𝑢))
3837simprbi 480 . . . . . . . . . . . . . . 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 793 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢 ∈ (𝐹𝑘))
41 simplrl 799 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑣𝑈)
4241adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑣𝑈)
4331, 42eqeltrrd 2699 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈)
44 rabid 3109 . . . . . . . . . . . . . . . 16 (𝑢 ∈ {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈} ↔ (𝑢 ∈ (𝐹𝑘) ∧ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈))
4540, 43, 44sylanbrc 697 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢 ∈ {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈})
4645, 9syl6eleqr 2709 . . . . . . . . . . . . . 14 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢𝐾)
47 elunii 4412 . . . . . . . . . . . . . 14 (((𝑓𝑘) ∈ 𝑢𝑢𝐾) → (𝑓𝑘) ∈ 𝐾)
4839, 46, 47syl2anc 692 . . . . . . . . . . . . 13 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → (𝑓𝑘) ∈ 𝐾)
4948rexlimdvaa 3026 . . . . . . . . . . . 12 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾))
5049expr 642 . . . . . . . . . . 11 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ 𝑘𝐴) → ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5150ralimdva 2957 . . . . . . . . . 10 (((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) → (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5251ex 450 . . . . . . . . 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 648 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ((𝑣𝑈𝑓𝑣) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5554imp 445 . . . . . 6 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾))
566adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑈 ⊆ ran 𝑆)
5756sselda 3587 . . . . . . . . 9 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ 𝑣𝑈) → 𝑣 ∈ ran 𝑆)
5857adantrr 752 . . . . . . . 8 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → 𝑣 ∈ ran 𝑆)
591rnmpt2 6730 . . . . . . . 8 ran 𝑆 = {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)}
6058, 59syl6eleq 2708 . . . . . . 7 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → 𝑣 ∈ {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)})
61 abid 2609 . . . . . . 7 (𝑣 ∈ {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)} ↔ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
6260, 61sylib 208 . . . . . 6 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
63 rexim 3003 . . . . . 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 3029 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
66 eldifn 3716 . . . . . . 7 ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ¬ (𝑓𝑘) ∈ 𝐾)
6766ralimi 2947 . . . . . 6 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾)
6867ad2antll 764 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾)
69 ralnex 2987 . . . . 5 (∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾 ↔ ¬ ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
7068, 69sylib 208 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ¬ ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
7165, 70pm2.65da 599 . . 3 (𝜑 → ¬ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
7271nexdv 1861 . 2 (𝜑 → ¬ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
7310, 72pm2.65i 185 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  {crab 2911  cdif 3556  cin 3558  wss 3559  𝒫 cpw 4135   cuni 4407  cmpt 4678  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  cmpt2 6612  Xcixp 7860  Fincfn 7907  cardccrd 8713  Compccmp 21112  UFLcufl 21627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-omul 7517  df-er 7694  df-map 7811  df-ixp 7861  df-en 7908  df-dom 7909  df-fin 7911  df-wdom 8416  df-card 8717  df-acn 8720  df-cmp 21113
This theorem is referenced by:  ptcmplem5  21783
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