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Theorem ptcmplem3 23558
Description: Lemma for ptcmp 23562. (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
ptcmplem3 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
Distinct variable groups:   𝑓,𝑘,𝑛,𝑢,𝑤,𝑧,𝐴   𝑓,𝐾,𝑢   𝑆,𝑘,𝑛,𝑢,𝑧   𝜑,𝑓,𝑘,𝑛,𝑢   𝑈,𝑘,𝑢,𝑧   𝑘,𝑉,𝑛,𝑢,𝑤,𝑧   𝑓,𝐹,𝑘,𝑛,𝑢,𝑤,𝑧   𝑓,𝑋,𝑘,𝑛,𝑢,𝑤,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝑆(𝑤,𝑓)   𝑈(𝑤,𝑓,𝑛)   𝐾(𝑧,𝑤,𝑘,𝑛)   𝑉(𝑓)

Proof of Theorem ptcmplem3
Dummy variables 𝑔 𝑚 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcmp.3 . . . 4 (𝜑𝐴𝑉)
2 rabexg 5332 . . . 4 (𝐴𝑉 → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ V)
31, 2syl 17 . . 3 (𝜑 → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ V)
4 ptcmp.1 . . . . 5 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
5 ptcmp.2 . . . . 5 𝑋 = X𝑛𝐴 (𝐹𝑛)
6 ptcmp.4 . . . . 5 (𝜑𝐹:𝐴⟶Comp)
7 ptcmp.5 . . . . 5 (𝜑𝑋 ∈ (UFL ∩ dom card))
8 ptcmplem2.5 . . . . 5 (𝜑𝑈 ⊆ ran 𝑆)
9 ptcmplem2.6 . . . . 5 (𝜑𝑋 = 𝑈)
10 ptcmplem2.7 . . . . 5 (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
114, 5, 1, 6, 7, 8, 9, 10ptcmplem2 23557 . . . 4 (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ dom card)
12 eldifi 4127 . . . . . . . 8 (𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾) → 𝑦 (𝐹𝑘))
13123ad2ant3 1136 . . . . . . 7 ((𝜑𝑦 ∈ V ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 (𝐹𝑘))
1413rabssdv 4073 . . . . . 6 (𝜑 → {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ⊆ (𝐹𝑘))
1514ralrimivw 3151 . . . . 5 (𝜑 → ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ⊆ (𝐹𝑘))
16 ss2iun 5016 . . . . 5 (∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ⊆ (𝐹𝑘) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ⊆ 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘))
1715, 16syl 17 . . . 4 (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ⊆ 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘))
18 ssnum 10034 . . . 4 (( 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ dom card ∧ 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ⊆ 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘)) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ∈ dom card)
1911, 17, 18syl2anc 585 . . 3 (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ∈ dom card)
20 elrabi 3678 . . . . 5 (𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} → 𝑘𝐴)
2110adantr 482 . . . . . . . 8 ((𝜑𝑘𝐴) → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
22 ssdif0 4364 . . . . . . . . 9 ( (𝐹𝑘) ⊆ 𝐾 ↔ ( (𝐹𝑘) ∖ 𝐾) = ∅)
236ffvelcdmda 7087 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ Comp)
2423adantr 482 . . . . . . . . . . . 12 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → (𝐹𝑘) ∈ Comp)
25 ptcmplem3.8 . . . . . . . . . . . . . 14 𝐾 = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈}
2625ssrab3 4081 . . . . . . . . . . . . 13 𝐾 ⊆ (𝐹𝑘)
2726a1i 11 . . . . . . . . . . . 12 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → 𝐾 ⊆ (𝐹𝑘))
28 simpr 486 . . . . . . . . . . . . 13 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → (𝐹𝑘) ⊆ 𝐾)
29 uniss 4917 . . . . . . . . . . . . . 14 (𝐾 ⊆ (𝐹𝑘) → 𝐾 (𝐹𝑘))
3026, 29mp1i 13 . . . . . . . . . . . . 13 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → 𝐾 (𝐹𝑘))
3128, 30eqssd 4000 . . . . . . . . . . . 12 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → (𝐹𝑘) = 𝐾)
32 eqid 2733 . . . . . . . . . . . . 13 (𝐹𝑘) = (𝐹𝑘)
3332cmpcov 22893 . . . . . . . . . . . 12 (((𝐹𝑘) ∈ Comp ∧ 𝐾 ⊆ (𝐹𝑘) ∧ (𝐹𝑘) = 𝐾) → ∃𝑡 ∈ (𝒫 𝐾 ∩ Fin) (𝐹𝑘) = 𝑡)
3424, 27, 31, 33syl3anc 1372 . . . . . . . . . . 11 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → ∃𝑡 ∈ (𝒫 𝐾 ∩ Fin) (𝐹𝑘) = 𝑡)
35 elfpw 9354 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ↔ (𝑡𝐾𝑡 ∈ Fin))
3635simplbi 499 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ (𝒫 𝐾 ∩ Fin) → 𝑡𝐾)
3736ad2antrl 727 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑡𝐾)
3837sselda 3983 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑥𝑡) → 𝑥𝐾)
39 imaeq2 6056 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑥 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
4039eleq1d 2819 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈))
4140, 25elrab2 3687 . . . . . . . . . . . . . . . . 17 (𝑥𝐾 ↔ (𝑥 ∈ (𝐹𝑘) ∧ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈))
4241simprbi 498 . . . . . . . . . . . . . . . 16 (𝑥𝐾 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈)
4338, 42syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑥𝑡) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈)
4443fmpttd 7115 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)):𝑡𝑈)
4544frnd 6726 . . . . . . . . . . . . 13 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ⊆ 𝑈)
4635simprbi 498 . . . . . . . . . . . . . . 15 (𝑡 ∈ (𝒫 𝐾 ∩ Fin) → 𝑡 ∈ Fin)
4746ad2antrl 727 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑡 ∈ Fin)
48 eqid 2733 . . . . . . . . . . . . . . . 16 (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) = (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
4948rnmpt 5955 . . . . . . . . . . . . . . 15 ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) = {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)}
50 abrexfi 9352 . . . . . . . . . . . . . . 15 (𝑡 ∈ Fin → {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)} ∈ Fin)
5149, 50eqeltrid 2838 . . . . . . . . . . . . . 14 (𝑡 ∈ Fin → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ Fin)
5247, 51syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ Fin)
53 elfpw 9354 . . . . . . . . . . . . 13 (ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ (𝒫 𝑈 ∩ Fin) ↔ (ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ⊆ 𝑈 ∧ ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ Fin))
5445, 52, 53sylanbrc 584 . . . . . . . . . . . 12 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ (𝒫 𝑈 ∩ Fin))
55 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
56 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
5756unieqd 4923 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
5855, 57eleq12d 2828 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → ((𝑓𝑛) ∈ (𝐹𝑛) ↔ (𝑓𝑘) ∈ (𝐹𝑘)))
59 simpr 486 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → 𝑓𝑋)
6059, 5eleqtrdi 2844 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → 𝑓X𝑛𝐴 (𝐹𝑛))
61 vex 3479 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑓 ∈ V
6261elixp 8898 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓X𝑛𝐴 (𝐹𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛)))
6362simprbi 498 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓X𝑛𝐴 (𝐹𝑛) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
6460, 63syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
65 simp-4r 783 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → 𝑘𝐴)
6658, 64, 65rspcdva 3614 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → (𝑓𝑘) ∈ (𝐹𝑘))
67 simplrr 777 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → (𝐹𝑘) = 𝑡)
6866, 67eleqtrd 2836 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → (𝑓𝑘) ∈ 𝑡)
69 eluni2 4913 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑘) ∈ 𝑡 ↔ ∃𝑥𝑡 (𝑓𝑘) ∈ 𝑥)
7068, 69sylib 217 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → ∃𝑥𝑡 (𝑓𝑘) ∈ 𝑥)
71 fveq1 6891 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑓 → (𝑤𝑘) = (𝑓𝑘))
7271eleq1d 2819 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑓 → ((𝑤𝑘) ∈ 𝑥 ↔ (𝑓𝑘) ∈ 𝑥))
73 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
7473mptpreima 6238 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑥}
7572, 74elrab2 3687 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ (𝑓𝑋 ∧ (𝑓𝑘) ∈ 𝑥))
7675baib 537 . . . . . . . . . . . . . . . . . . . 20 (𝑓𝑋 → (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ (𝑓𝑘) ∈ 𝑥))
7776ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) ∧ 𝑥𝑡) → (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ (𝑓𝑘) ∈ 𝑥))
7877rexbidva 3177 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → (∃𝑥𝑡 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ ∃𝑥𝑡 (𝑓𝑘) ∈ 𝑥))
7970, 78mpbird 257 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → ∃𝑥𝑡 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
80 eliun 5002 . . . . . . . . . . . . . . . . 17 (𝑓 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ ∃𝑥𝑡 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
8179, 80sylibr 233 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → 𝑓 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
8281ex 414 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → (𝑓𝑋𝑓 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
8382ssrdv 3989 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑋 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
8443ralrimiva 3147 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ∀𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈)
85 dfiun2g 5034 . . . . . . . . . . . . . . . 16 (∀𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) = {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)})
8684, 85syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) = {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)})
8749unieqi 4922 . . . . . . . . . . . . . . 15 ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) = {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)}
8886, 87eqtr4di 2791 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
8983, 88sseqtrd 4023 . . . . . . . . . . . . 13 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑋 ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
9045unissd 4919 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ⊆ 𝑈)
919ad3antrrr 729 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑋 = 𝑈)
9290, 91sseqtrrd 4024 . . . . . . . . . . . . 13 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ⊆ 𝑋)
9389, 92eqssd 4000 . . . . . . . . . . . 12 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑋 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
94 unieq 4920 . . . . . . . . . . . . 13 (𝑧 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) → 𝑧 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
9594rspceeqv 3634 . . . . . . . . . . . 12 ((ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑋 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
9654, 93, 95syl2anc 585 . . . . . . . . . . 11 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
9734, 96rexlimddv 3162 . . . . . . . . . 10 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
9897ex 414 . . . . . . . . 9 ((𝜑𝑘𝐴) → ( (𝐹𝑘) ⊆ 𝐾 → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧))
9922, 98biimtrrid 242 . . . . . . . 8 ((𝜑𝑘𝐴) → (( (𝐹𝑘) ∖ 𝐾) = ∅ → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧))
10021, 99mtod 197 . . . . . . 7 ((𝜑𝑘𝐴) → ¬ ( (𝐹𝑘) ∖ 𝐾) = ∅)
101 neq0 4346 . . . . . . 7 (¬ ( (𝐹𝑘) ∖ 𝐾) = ∅ ↔ ∃𝑦 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
102100, 101sylib 217 . . . . . 6 ((𝜑𝑘𝐴) → ∃𝑦 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
103 rexv 3500 . . . . . 6 (∃𝑦 ∈ V 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ ∃𝑦 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
104102, 103sylibr 233 . . . . 5 ((𝜑𝑘𝐴) → ∃𝑦 ∈ V 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
10520, 104sylan2 594 . . . 4 ((𝜑𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}) → ∃𝑦 ∈ V 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
106105ralrimiva 3147 . . 3 (𝜑 → ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}∃𝑦 ∈ V 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
107 eleq1 2822 . . . 4 (𝑦 = (𝑔𝑘) → (𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
108107ac6num 10474 . . 3 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ V ∧ 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ∈ dom card ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}∃𝑦 ∈ V 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → ∃𝑔(𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
1093, 19, 106, 108syl3anc 1372 . 2 (𝜑 → ∃𝑔(𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
1101adantr 482 . . . 4 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝐴𝑉)
111110mptexd 7226 . . 3 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) ∈ V)
112 fvex 6905 . . . . . . . 8 (𝐹𝑚) ∈ V
113112uniex 7731 . . . . . . 7 (𝐹𝑚) ∈ V
114113uniex 7731 . . . . . 6 (𝐹𝑚) ∈ V
115 fvex 6905 . . . . . 6 (𝑔𝑚) ∈ V
116114, 115ifex 4579 . . . . 5 if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚)) ∈ V
117116rgenw 3066 . . . 4 𝑚𝐴 if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚)) ∈ V
118 eqid 2733 . . . . 5 (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚)))
119118fnmpt 6691 . . . 4 (∀𝑚𝐴 if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚)) ∈ V → (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) Fn 𝐴)
120117, 119mp1i 13 . . 3 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) Fn 𝐴)
12157breq1d 5159 . . . . . . 7 (𝑛 = 𝑘 → ( (𝐹𝑛) ≈ 1o (𝐹𝑘) ≈ 1o))
122121notbid 318 . . . . . 6 (𝑛 = 𝑘 → (¬ (𝐹𝑛) ≈ 1o ↔ ¬ (𝐹𝑘) ≈ 1o))
123122ralrab 3690 . . . . 5 (∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ ∀𝑘𝐴 (𝐹𝑘) ≈ 1o → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
124 iftrue 4535 . . . . . . . . . . 11 ( (𝐹𝑘) ≈ 1o → if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) = (𝐹𝑘))
125124ad2antll 728 . . . . . . . . . 10 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V) ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) → if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) = (𝐹𝑘))
126102adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) → ∃𝑦 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
12712adantl 483 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 (𝐹𝑘))
128 simplrr 777 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → (𝐹𝑘) ≈ 1o)
129 en1b 9023 . . . . . . . . . . . . . . . 16 ( (𝐹𝑘) ≈ 1o (𝐹𝑘) = { (𝐹𝑘)})
130128, 129sylib 217 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → (𝐹𝑘) = { (𝐹𝑘)})
131127, 130eleqtrd 2836 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 ∈ { (𝐹𝑘)})
132 elsni 4646 . . . . . . . . . . . . . 14 (𝑦 ∈ { (𝐹𝑘)} → 𝑦 = (𝐹𝑘))
133131, 132syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 = (𝐹𝑘))
134 simpr 486 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
135133, 134eqeltrrd 2835 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → (𝐹𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))
136126, 135exlimddv 1939 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) → (𝐹𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))
137136adantlr 714 . . . . . . . . . 10 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V) ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) → (𝐹𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))
138125, 137eqeltrd 2834 . . . . . . . . 9 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V) ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) → if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾))
139138a1d 25 . . . . . . . 8 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V) ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1o)) → ((¬ (𝐹𝑘) ≈ 1o → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
140139expr 458 . . . . . . 7 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V) ∧ 𝑘𝐴) → ( (𝐹𝑘) ≈ 1o → ((¬ (𝐹𝑘) ≈ 1o → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾))))
141 pm2.27 42 . . . . . . . 8 (𝐹𝑘) ≈ 1o → ((¬ (𝐹𝑘) ≈ 1o → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
142 iffalse 4538 . . . . . . . . 9 (𝐹𝑘) ≈ 1o → if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) = (𝑔𝑘))
143142eleq1d 2819 . . . . . . . 8 (𝐹𝑘) ≈ 1o → (if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
144141, 143sylibrd 259 . . . . . . 7 (𝐹𝑘) ≈ 1o → ((¬ (𝐹𝑘) ≈ 1o → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
145140, 144pm2.61d1 180 . . . . . 6 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V) ∧ 𝑘𝐴) → ((¬ (𝐹𝑘) ≈ 1o → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
146145ralimdva 3168 . . . . 5 ((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V) → (∀𝑘𝐴 (𝐹𝑘) ≈ 1o → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
147123, 146biimtrid 241 . . . 4 ((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V) → (∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
148147impr 456 . . 3 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾))
149 fneq1 6641 . . . . . 6 (𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) → (𝑓 Fn 𝐴 ↔ (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) Fn 𝐴))
150 fveq1 6891 . . . . . . . . 9 (𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) → (𝑓𝑘) = ((𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚)))‘𝑘))
151 fveq2 6892 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
152151unieqd 4923 . . . . . . . . . . . 12 (𝑚 = 𝑘 (𝐹𝑚) = (𝐹𝑘))
153152breq1d 5159 . . . . . . . . . . 11 (𝑚 = 𝑘 → ( (𝐹𝑚) ≈ 1o (𝐹𝑘) ≈ 1o))
154152unieqd 4923 . . . . . . . . . . 11 (𝑚 = 𝑘 (𝐹𝑚) = (𝐹𝑘))
155 fveq2 6892 . . . . . . . . . . 11 (𝑚 = 𝑘 → (𝑔𝑚) = (𝑔𝑘))
156153, 154, 155ifbieq12d 4557 . . . . . . . . . 10 (𝑚 = 𝑘 → if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚)) = if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)))
157 fvex 6905 . . . . . . . . . . . . 13 (𝐹𝑘) ∈ V
158157uniex 7731 . . . . . . . . . . . 12 (𝐹𝑘) ∈ V
159158uniex 7731 . . . . . . . . . . 11 (𝐹𝑘) ∈ V
160 fvex 6905 . . . . . . . . . . 11 (𝑔𝑘) ∈ V
161159, 160ifex 4579 . . . . . . . . . 10 if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ V
162156, 118, 161fvmpt 6999 . . . . . . . . 9 (𝑘𝐴 → ((𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚)))‘𝑘) = if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)))
163150, 162sylan9eq 2793 . . . . . . . 8 ((𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) ∧ 𝑘𝐴) → (𝑓𝑘) = if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)))
164163eleq1d 2819 . . . . . . 7 ((𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) ∧ 𝑘𝐴) → ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
165164ralbidva 3176 . . . . . 6 (𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) → (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
166149, 165anbi12d 632 . . . . 5 (𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) → ((𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) ↔ ((𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) Fn 𝐴 ∧ ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾))))
167166spcegv 3588 . . . 4 ((𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) ∈ V → (((𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) Fn 𝐴 ∧ ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))))
1681673impib 1117 . . 3 (((𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) ∈ V ∧ (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1o, (𝐹𝑚), (𝑔𝑚))) Fn 𝐴 ∧ ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1o, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
169111, 120, 148, 168syl3anc 1372 . 2 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
170109, 169exlimddv 1939 1 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  cdif 3946  cin 3948  wss 3949  c0 4323  ifcif 4529  𝒫 cpw 4603  {csn 4629   cuni 4909   ciun 4998   class class class wbr 5149  cmpt 5232  ccnv 5676  dom cdm 5677  ran crn 5678  cima 5680   Fn wfn 6539  wf 6540  cfv 6544  cmpo 7411  1oc1o 8459  Xcixp 8891  cen 8936  Fincfn 8939  cardccrd 9930  Compccmp 22890  UFLcufl 23404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-fin 8943  df-wdom 9560  df-card 9934  df-acn 9937  df-cmp 22891
This theorem is referenced by:  ptcmplem4  23559
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