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Theorem ptcmplem4 24050
Description: Lemma for ptcmp 24053. (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 24049 . 2 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
11 simprl 769 . . . . . . . . 9 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓 Fn 𝐴)
12 eldifi 4126 . . . . . . . . . . . 12 ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → (𝑓𝑘) ∈ (𝐹𝑘))
1312ralimi 3073 . . . . . . . . . . 11 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
14 fveq2 6901 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
15 fveq2 6901 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
1615unieqd 4926 . . . . . . . . . . . . 13 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
1714, 16eleq12d 2820 . . . . . . . . . . . 12 (𝑛 = 𝑘 → ((𝑓𝑛) ∈ (𝐹𝑛) ↔ (𝑓𝑘) ∈ (𝐹𝑘)))
1817cbvralvw 3225 . . . . . . . . . . 11 (∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛) ↔ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
1913, 18sylibr 233 . . . . . . . . . 10 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
2019ad2antll 727 . . . . . . . . 9 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
21 vex 3466 . . . . . . . . . 10 𝑓 ∈ V
2221elixp 8933 . . . . . . . . 9 (𝑓X𝑛𝐴 (𝐹𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛)))
2311, 20, 22sylanbrc 581 . . . . . . . 8 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓X𝑛𝐴 (𝐹𝑛))
2423, 2eleqtrrdi 2837 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓𝑋)
257adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑋 = 𝑈)
2624, 25eleqtrd 2828 . . . . . 6 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓 𝑈)
27 eluni2 4917 . . . . . 6 (𝑓 𝑈 ↔ ∃𝑣𝑈 𝑓𝑣)
2826, 27sylib 217 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∃𝑣𝑈 𝑓𝑣)
29 simplrr 776 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓𝑣)
3029adantr 479 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑓𝑣)
31 simprr 771 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
3230, 31eleqtrd 2828 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
33 fveq1 6900 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑓 → (𝑤𝑘) = (𝑓𝑘))
3433eleq1d 2811 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑓 → ((𝑤𝑘) ∈ 𝑢 ↔ (𝑓𝑘) ∈ 𝑢))
35 eqid 2726 . . . . . . . . . . . . . . . . . 18 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
3635mptpreima 6249 . . . . . . . . . . . . . . . . 17 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢}
3734, 36elrab2 3684 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑓𝑋 ∧ (𝑓𝑘) ∈ 𝑢))
3837simprbi 495 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢 ∈ (𝐹𝑘))
41 simplrl 775 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑣𝑈)
4241adantr 479 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑣𝑈)
4331, 42eqeltrrd 2827 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈)
44 rabid 3440 . . . . . . . . . . . . . . . 16 (𝑢 ∈ {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈} ↔ (𝑢 ∈ (𝐹𝑘) ∧ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈))
4540, 43, 44sylanbrc 581 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢 ∈ {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈})
4645, 9eleqtrrdi 2837 . . . . . . . . . . . . . 14 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢𝐾)
47 elunii 4918 . . . . . . . . . . . . . 14 (((𝑓𝑘) ∈ 𝑢𝑢𝐾) → (𝑓𝑘) ∈ 𝐾)
4839, 46, 47syl2anc 582 . . . . . . . . . . . . 13 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → (𝑓𝑘) ∈ 𝐾)
4948rexlimdvaa 3146 . . . . . . . . . . . 12 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾))
5049expr 455 . . . . . . . . . . 11 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ 𝑘𝐴) → ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5150ralimdva 3157 . . . . . . . . . 10 (((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) → (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5251ex 411 . . . . . . . . 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 453 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ((𝑣𝑈𝑓𝑣) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5554imp 405 . . . . . 6 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾))
566adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑈 ⊆ ran 𝑆)
5756sselda 3979 . . . . . . . . 9 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ 𝑣𝑈) → 𝑣 ∈ ran 𝑆)
5857adantrr 715 . . . . . . . 8 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → 𝑣 ∈ ran 𝑆)
591rnmpo 7559 . . . . . . . 8 ran 𝑆 = {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)}
6058, 59eleqtrdi 2836 . . . . . . 7 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → 𝑣 ∈ {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)})
61 abid 2707 . . . . . . 7 (𝑣 ∈ {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)} ↔ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
6260, 61sylib 217 . . . . . 6 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
63 rexim 3077 . . . . . 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 3151 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
66 eldifn 4127 . . . . . . 7 ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ¬ (𝑓𝑘) ∈ 𝐾)
6766ralimi 3073 . . . . . 6 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾)
6867ad2antll 727 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾)
69 ralnex 3062 . . . . 5 (∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾 ↔ ¬ ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
7068, 69sylib 217 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ¬ ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
7165, 70pm2.65da 815 . . 3 (𝜑 → ¬ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
7271nexdv 1932 . 2 (𝜑 → ¬ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
7310, 72pm2.65i 193 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wral 3051  wrex 3060  {crab 3419  cdif 3944  cin 3946  wss 3947  𝒫 cpw 4607   cuni 4913  cmpt 5236  ccnv 5681  dom cdm 5682  ran crn 5683  cima 5685   Fn wfn 6549  wf 6550  cfv 6554  cmpo 7426  Xcixp 8926  Fincfn 8974  cardccrd 9978  Compccmp 23381  UFLcufl 23895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-oadd 8500  df-omul 8501  df-er 8734  df-map 8857  df-ixp 8927  df-en 8975  df-dom 8976  df-fin 8978  df-wdom 9608  df-card 9982  df-acn 9985  df-cmp 23382
This theorem is referenced by:  ptcmplem5  24051
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