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Theorem ptcmplem4 23406
Description: Lemma for ptcmp 23409. (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 23405 . 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 4086 . . . . . . . . . . . 12 ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → (𝑓𝑘) ∈ (𝐹𝑘))
1312ralimi 3086 . . . . . . . . . . 11 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
14 fveq2 6842 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
15 fveq2 6842 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
1615unieqd 4879 . . . . . . . . . . . . 13 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
1714, 16eleq12d 2832 . . . . . . . . . . . 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 3449 . . . . . . . . . 10 𝑓 ∈ V
2221elixp 8842 . . . . . . . . 9 (𝑓X𝑛𝐴 (𝐹𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛)))
2311, 20, 22sylanbrc 583 . . . . . . . 8 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓X𝑛𝐴 (𝐹𝑛))
2423, 2eleqtrrdi 2849 . . . . . . 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 2840 . . . . . 6 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝑓 𝑈)
27 eluni2 4869 . . . . . 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 481 . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
33 fveq1 6841 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑓 → (𝑤𝑘) = (𝑓𝑘))
3433eleq1d 2822 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑓 → ((𝑤𝑘) ∈ 𝑢 ↔ (𝑓𝑘) ∈ 𝑢))
35 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
3635mptpreima 6190 . . . . . . . . . . . . . . . . 17 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢}
3734, 36elrab2 3648 . . . . . . . . . . . . . . . 16 (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑓𝑋 ∧ (𝑓𝑘) ∈ 𝑢))
3837simprbi 497 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑣𝑈)
4331, 42eqeltrrd 2839 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈)
44 rabid 3427 . . . . . . . . . . . . . . . 16 (𝑢 ∈ {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈} ↔ (𝑢 ∈ (𝐹𝑘) ∧ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈))
4540, 43, 44sylanbrc 583 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢 ∈ {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈})
4645, 9eleqtrrdi 2849 . . . . . . . . . . . . . 14 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → 𝑢𝐾)
47 elunii 4870 . . . . . . . . . . . . . 14 (((𝑓𝑘) ∈ 𝑢𝑢𝐾) → (𝑓𝑘) ∈ 𝐾)
4839, 46, 47syl2anc 584 . . . . . . . . . . . . 13 (((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑢 ∈ (𝐹𝑘) ∧ 𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))) → (𝑓𝑘) ∈ 𝐾)
4948rexlimdvaa 3153 . . . . . . . . . . . 12 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ (𝑘𝐴 ∧ (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾))
5049expr 457 . . . . . . . . . . 11 ((((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) ∧ 𝑘𝐴) → ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5150ralimdva 3164 . . . . . . . . . 10 (((𝜑𝑓 Fn 𝐴) ∧ (𝑣𝑈𝑓𝑣)) → (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5251ex 413 . . . . . . . . 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 455 . . . . . . 7 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ((𝑣𝑈𝑓𝑣) → ∀𝑘𝐴 (∃𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) → (𝑓𝑘) ∈ 𝐾)))
5554imp 407 . . . . . 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 3944 . . . . . . . . 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 7489 . . . . . . . 8 ran 𝑆 = {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)}
6058, 59eleqtrdi 2848 . . . . . . 7 (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) ∧ (𝑣𝑈𝑓𝑣)) → 𝑣 ∈ {𝑣 ∣ ∃𝑘𝐴𝑢 ∈ (𝐹𝑘)𝑣 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)})
61 abid 2717 . . . . . . 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 3090 . . . . . 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 3158 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∃𝑘𝐴 (𝑓𝑘) ∈ 𝐾)
66 eldifn 4087 . . . . . . 7 ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ¬ (𝑓𝑘) ∈ 𝐾)
6766ralimi 3086 . . . . . 6 (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾)
6867ad2antll 727 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∀𝑘𝐴 ¬ (𝑓𝑘) ∈ 𝐾)
69 ralnex 3075 . . . . 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 1939 . 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 396   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  wrex 3073  {crab 3407  cdif 3907  cin 3909  wss 3910  𝒫 cpw 4560   cuni 4865  cmpt 5188  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  cmpo 7359  Xcixp 8835  Fincfn 8883  cardccrd 9871  Compccmp 22737  UFLcufl 23251
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-wdom 9501  df-card 9875  df-acn 9878  df-cmp 22738
This theorem is referenced by:  ptcmplem5  23407
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