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Theorem ptcmplem3 21763
Description: Lemma for ptcmp 21767. (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 4777 . . . 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 21762 . . . 4 (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card)
12 eldifi 3715 . . . . . . . 8 (𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾) → 𝑦 (𝐹𝑘))
13123ad2ant3 1082 . . . . . . 7 ((𝜑𝑦 ∈ V ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 (𝐹𝑘))
1413rabssdv 3666 . . . . . 6 (𝜑 → {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ⊆ (𝐹𝑘))
1514ralrimivw 2966 . . . . 5 (𝜑 → ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ⊆ (𝐹𝑘))
16 ss2iun 4507 . . . . 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 8807 . . . 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 692 . . 3 (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} {𝑦 ∈ V ∣ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)} ∈ dom card)
20 elrabi 3347 . . . . 5 (𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} → 𝑘𝐴)
2110adantr 481 . . . . . . . 8 ((𝜑𝑘𝐴) → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
22 ssdif0 3921 . . . . . . . . 9 ( (𝐹𝑘) ⊆ 𝐾 ↔ ( (𝐹𝑘) ∖ 𝐾) = ∅)
236ffvelrnda 6316 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ Comp)
2423adantr 481 . . . . . . . . . . . 12 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → (𝐹𝑘) ∈ Comp)
25 ptcmplem3.8 . . . . . . . . . . . . . 14 𝐾 = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈}
26 ssrab2 3671 . . . . . . . . . . . . . 14 {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈} ⊆ (𝐹𝑘)
2725, 26eqsstri 3619 . . . . . . . . . . . . 13 𝐾 ⊆ (𝐹𝑘)
2827a1i 11 . . . . . . . . . . . 12 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → 𝐾 ⊆ (𝐹𝑘))
29 simpr 477 . . . . . . . . . . . . 13 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → (𝐹𝑘) ⊆ 𝐾)
30 uniss 4429 . . . . . . . . . . . . . 14 (𝐾 ⊆ (𝐹𝑘) → 𝐾 (𝐹𝑘))
3127, 30mp1i 13 . . . . . . . . . . . . 13 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → 𝐾 (𝐹𝑘))
3229, 31eqssd 3605 . . . . . . . . . . . 12 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → (𝐹𝑘) = 𝐾)
33 eqid 2626 . . . . . . . . . . . . 13 (𝐹𝑘) = (𝐹𝑘)
3433cmpcov 21097 . . . . . . . . . . . 12 (((𝐹𝑘) ∈ Comp ∧ 𝐾 ⊆ (𝐹𝑘) ∧ (𝐹𝑘) = 𝐾) → ∃𝑡 ∈ (𝒫 𝐾 ∩ Fin) (𝐹𝑘) = 𝑡)
3524, 28, 32, 34syl3anc 1323 . . . . . . . . . . 11 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → ∃𝑡 ∈ (𝒫 𝐾 ∩ Fin) (𝐹𝑘) = 𝑡)
36 elfpw 8213 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ↔ (𝑡𝐾𝑡 ∈ Fin))
3736simplbi 476 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ (𝒫 𝐾 ∩ Fin) → 𝑡𝐾)
3837ad2antrl 763 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑡𝐾)
3938sselda 3588 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑥𝑡) → 𝑥𝐾)
40 imaeq2 5425 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑥 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
4140eleq1d 2688 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑥 → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈))
4241, 25elrab2 3353 . . . . . . . . . . . . . . . . 17 (𝑥𝐾 ↔ (𝑥 ∈ (𝐹𝑘) ∧ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈))
4342simprbi 480 . . . . . . . . . . . . . . . 16 (𝑥𝐾 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈)
4439, 43syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑥𝑡) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈)
45 eqid 2626 . . . . . . . . . . . . . . 15 (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) = (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
4644, 45fmptd 6341 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)):𝑡𝑈)
47 frn 6012 . . . . . . . . . . . . . 14 ((𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)):𝑡𝑈 → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ⊆ 𝑈)
4846, 47syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ⊆ 𝑈)
4936simprbi 480 . . . . . . . . . . . . . . 15 (𝑡 ∈ (𝒫 𝐾 ∩ Fin) → 𝑡 ∈ Fin)
5049ad2antrl 763 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑡 ∈ Fin)
5145rnmpt 5335 . . . . . . . . . . . . . . 15 ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) = {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)}
52 abrexfi 8211 . . . . . . . . . . . . . . 15 (𝑡 ∈ Fin → {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)} ∈ Fin)
5351, 52syl5eqel 2708 . . . . . . . . . . . . . 14 (𝑡 ∈ Fin → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ Fin)
5450, 53syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ Fin)
55 elfpw 8213 . . . . . . . . . . . . 13 (ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ (𝒫 𝑈 ∩ Fin) ↔ (ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ⊆ 𝑈 ∧ ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ Fin))
5648, 54, 55sylanbrc 697 . . . . . . . . . . . 12 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ (𝒫 𝑈 ∩ Fin))
57 simp-4r 806 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → 𝑘𝐴)
58 simpr 477 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → 𝑓𝑋)
5958, 5syl6eleq 2714 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → 𝑓X𝑛𝐴 (𝐹𝑛))
60 vex 3194 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑓 ∈ V
6160elixp 7860 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓X𝑛𝐴 (𝐹𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛)))
6261simprbi 480 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓X𝑛𝐴 (𝐹𝑛) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
6359, 62syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
64 fveq2 6150 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑘 → (𝑓𝑛) = (𝑓𝑘))
65 fveq2 6150 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
6665unieqd 4417 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
6764, 66eleq12d 2698 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → ((𝑓𝑛) ∈ (𝐹𝑛) ↔ (𝑓𝑘) ∈ (𝐹𝑘)))
6867rspcv 3296 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝐴 → (∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛) → (𝑓𝑘) ∈ (𝐹𝑘)))
6957, 63, 68sylc 65 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → (𝑓𝑘) ∈ (𝐹𝑘))
70 simplrr 800 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → (𝐹𝑘) = 𝑡)
7169, 70eleqtrd 2706 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → (𝑓𝑘) ∈ 𝑡)
72 eluni2 4411 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑘) ∈ 𝑡 ↔ ∃𝑥𝑡 (𝑓𝑘) ∈ 𝑥)
7371, 72sylib 208 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → ∃𝑥𝑡 (𝑓𝑘) ∈ 𝑥)
74 fveq1 6149 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑓 → (𝑤𝑘) = (𝑓𝑘))
7574eleq1d 2688 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑓 → ((𝑤𝑘) ∈ 𝑥 ↔ (𝑓𝑘) ∈ 𝑥))
76 eqid 2626 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
7776mptpreima 5590 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑥}
7875, 77elrab2 3353 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ (𝑓𝑋 ∧ (𝑓𝑘) ∈ 𝑥))
7978baib 943 . . . . . . . . . . . . . . . . . . . 20 (𝑓𝑋 → (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ (𝑓𝑘) ∈ 𝑥))
8079ad2antlr 762 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) ∧ 𝑥𝑡) → (𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ (𝑓𝑘) ∈ 𝑥))
8180rexbidva 3047 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → (∃𝑥𝑡 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ ∃𝑥𝑡 (𝑓𝑘) ∈ 𝑥))
8273, 81mpbird 247 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → ∃𝑥𝑡 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
83 eliun 4495 . . . . . . . . . . . . . . . . 17 (𝑓 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ↔ ∃𝑥𝑡 𝑓 ∈ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
8482, 83sylibr 224 . . . . . . . . . . . . . . . 16 (((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) ∧ 𝑓𝑋) → 𝑓 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
8584ex 450 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → (𝑓𝑋𝑓 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
8685ssrdv 3594 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑋 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))
8744ralrimiva 2965 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ∀𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈)
88 dfiun2g 4523 . . . . . . . . . . . . . . . 16 (∀𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) ∈ 𝑈 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) = {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)})
8987, 88syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) = {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)})
9051unieqi 4416 . . . . . . . . . . . . . . 15 ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) = {𝑓 ∣ ∃𝑥𝑡 𝑓 = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)}
9189, 90syl6eqr 2678 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑥𝑡 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥) = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
9286, 91sseqtrd 3625 . . . . . . . . . . . . 13 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑋 ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
9348unissd 4433 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ⊆ 𝑈)
949ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑋 = 𝑈)
9593, 94sseqtr4d 3626 . . . . . . . . . . . . 13 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ⊆ 𝑋)
9692, 95eqssd 3605 . . . . . . . . . . . 12 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → 𝑋 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
97 unieq 4415 . . . . . . . . . . . . . 14 (𝑧 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) → 𝑧 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)))
9897eqeq2d 2636 . . . . . . . . . . . . 13 (𝑧 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) → (𝑋 = 𝑧𝑋 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))))
9998rspcev 3300 . . . . . . . . . . . 12 ((ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥)) ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑋 = ran (𝑥𝑡 ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑥))) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
10056, 96, 99syl2anc 692 . . . . . . . . . . 11 ((((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) ∧ (𝑡 ∈ (𝒫 𝐾 ∩ Fin) ∧ (𝐹𝑘) = 𝑡)) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
10135, 100rexlimddv 3033 . . . . . . . . . 10 (((𝜑𝑘𝐴) ∧ (𝐹𝑘) ⊆ 𝐾) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
102101ex 450 . . . . . . . . 9 ((𝜑𝑘𝐴) → ( (𝐹𝑘) ⊆ 𝐾 → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧))
10322, 102syl5bir 233 . . . . . . . 8 ((𝜑𝑘𝐴) → (( (𝐹𝑘) ∖ 𝐾) = ∅ → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧))
10421, 103mtod 189 . . . . . . 7 ((𝜑𝑘𝐴) → ¬ ( (𝐹𝑘) ∖ 𝐾) = ∅)
105 neq0 3911 . . . . . . 7 (¬ ( (𝐹𝑘) ∖ 𝐾) = ∅ ↔ ∃𝑦 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
106104, 105sylib 208 . . . . . 6 ((𝜑𝑘𝐴) → ∃𝑦 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
107 rexv 3211 . . . . . 6 (∃𝑦 ∈ V 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ ∃𝑦 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
108106, 107sylibr 224 . . . . 5 ((𝜑𝑘𝐴) → ∃𝑦 ∈ V 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
10920, 108sylan2 491 . . . 4 ((𝜑𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}) → ∃𝑦 ∈ V 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
110109ralrimiva 2965 . . 3 (𝜑 → ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}∃𝑦 ∈ V 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
111 eleq1 2692 . . . 4 (𝑦 = (𝑔𝑘) → (𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
112111ac6num 9246 . . 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)))
1133, 19, 110, 112syl3anc 1323 . 2 (𝜑 → ∃𝑔(𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
1141adantr 481 . . . 4 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → 𝐴𝑉)
115 mptexg 6439 . . . 4 (𝐴𝑉 → (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) ∈ V)
116114, 115syl 17 . . 3 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) ∈ V)
117 fvex 6160 . . . . . . . 8 (𝐹𝑚) ∈ V
118117uniex 6907 . . . . . . 7 (𝐹𝑚) ∈ V
119118uniex 6907 . . . . . 6 (𝐹𝑚) ∈ V
120 fvex 6160 . . . . . 6 (𝑔𝑚) ∈ V
121119, 120ifex 4133 . . . . 5 if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚)) ∈ V
122121rgenw 2924 . . . 4 𝑚𝐴 if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚)) ∈ V
123 eqid 2626 . . . . 5 (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚)))
124123fnmpt 5979 . . . 4 (∀𝑚𝐴 if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚)) ∈ V → (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) Fn 𝐴)
125122, 124mp1i 13 . . 3 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) Fn 𝐴)
12666breq1d 4628 . . . . . . 7 (𝑛 = 𝑘 → ( (𝐹𝑛) ≈ 1𝑜 (𝐹𝑘) ≈ 1𝑜))
127126notbid 308 . . . . . 6 (𝑛 = 𝑘 → (¬ (𝐹𝑛) ≈ 1𝑜 ↔ ¬ (𝐹𝑘) ≈ 1𝑜))
128127ralrab 3355 . . . . 5 (∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ ∀𝑘𝐴 (𝐹𝑘) ≈ 1𝑜 → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
129 iftrue 4069 . . . . . . . . . . 11 ( (𝐹𝑘) ≈ 1𝑜 → if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) = (𝐹𝑘))
130129ad2antll 764 . . . . . . . . . 10 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V) ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) → if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) = (𝐹𝑘))
131106adantrr 752 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) → ∃𝑦 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
13212adantl 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 (𝐹𝑘))
133 simplrr 800 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → (𝐹𝑘) ≈ 1𝑜)
134 en1b 7969 . . . . . . . . . . . . . . . 16 ( (𝐹𝑘) ≈ 1𝑜 (𝐹𝑘) = { (𝐹𝑘)})
135133, 134sylib 208 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → (𝐹𝑘) = { (𝐹𝑘)})
136132, 135eleqtrd 2706 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 ∈ { (𝐹𝑘)})
137 elsni 4170 . . . . . . . . . . . . . 14 (𝑦 ∈ { (𝐹𝑘)} → 𝑦 = (𝐹𝑘))
138136, 137syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 = (𝐹𝑘))
139 simpr 477 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾))
140138, 139eqeltrrd 2705 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) ∧ 𝑦 ∈ ( (𝐹𝑘) ∖ 𝐾)) → (𝐹𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))
141131, 140exlimddv 1865 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) → (𝐹𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))
142141adantlr 750 . . . . . . . . . 10 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V) ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) → (𝐹𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))
143130, 142eqeltrd 2704 . . . . . . . . 9 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V) ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) → if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾))
144143a1d 25 . . . . . . . 8 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V) ∧ (𝑘𝐴 (𝐹𝑘) ≈ 1𝑜)) → ((¬ (𝐹𝑘) ≈ 1𝑜 → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
145144expr 642 . . . . . . 7 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V) ∧ 𝑘𝐴) → ( (𝐹𝑘) ≈ 1𝑜 → ((¬ (𝐹𝑘) ≈ 1𝑜 → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾))))
146 pm2.27 42 . . . . . . . 8 (𝐹𝑘) ≈ 1𝑜 → ((¬ (𝐹𝑘) ≈ 1𝑜 → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
147 iffalse 4072 . . . . . . . . 9 (𝐹𝑘) ≈ 1𝑜 → if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) = (𝑔𝑘))
148147eleq1d 2688 . . . . . . . 8 (𝐹𝑘) ≈ 1𝑜 → (if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
149146, 148sylibrd 249 . . . . . . 7 (𝐹𝑘) ≈ 1𝑜 → ((¬ (𝐹𝑘) ≈ 1𝑜 → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
150145, 149pm2.61d1 171 . . . . . 6 (((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V) ∧ 𝑘𝐴) → ((¬ (𝐹𝑘) ≈ 1𝑜 → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
151150ralimdva 2961 . . . . 5 ((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V) → (∀𝑘𝐴 (𝐹𝑘) ≈ 1𝑜 → (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)) → ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
152128, 151syl5bi 232 . . . 4 ((𝜑𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V) → (∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) → ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
153152impr 648 . . 3 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾))
154 fneq1 5939 . . . . . 6 (𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) → (𝑓 Fn 𝐴 ↔ (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) Fn 𝐴))
155 fveq1 6149 . . . . . . . . 9 (𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) → (𝑓𝑘) = ((𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚)))‘𝑘))
156 fveq2 6150 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
157156unieqd 4417 . . . . . . . . . . . 12 (𝑚 = 𝑘 (𝐹𝑚) = (𝐹𝑘))
158157breq1d 4628 . . . . . . . . . . 11 (𝑚 = 𝑘 → ( (𝐹𝑚) ≈ 1𝑜 (𝐹𝑘) ≈ 1𝑜))
159157unieqd 4417 . . . . . . . . . . 11 (𝑚 = 𝑘 (𝐹𝑚) = (𝐹𝑘))
160 fveq2 6150 . . . . . . . . . . 11 (𝑚 = 𝑘 → (𝑔𝑚) = (𝑔𝑘))
161158, 159, 160ifbieq12d 4090 . . . . . . . . . 10 (𝑚 = 𝑘 → if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚)) = if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)))
162 fvex 6160 . . . . . . . . . . . . 13 (𝐹𝑘) ∈ V
163162uniex 6907 . . . . . . . . . . . 12 (𝐹𝑘) ∈ V
164163uniex 6907 . . . . . . . . . . 11 (𝐹𝑘) ∈ V
165 fvex 6160 . . . . . . . . . . 11 (𝑔𝑘) ∈ V
166164, 165ifex 4133 . . . . . . . . . 10 if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ V
167161, 123, 166fvmpt 6240 . . . . . . . . 9 (𝑘𝐴 → ((𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚)))‘𝑘) = if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)))
168155, 167sylan9eq 2680 . . . . . . . 8 ((𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) ∧ 𝑘𝐴) → (𝑓𝑘) = if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)))
169168eleq1d 2688 . . . . . . 7 ((𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) ∧ 𝑘𝐴) → ((𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
170169ralbidva 2984 . . . . . 6 (𝑓 = (𝑚𝐴 ↦ if( (𝐹𝑚) ≈ 1𝑜, (𝐹𝑚), (𝑔𝑚))) → (∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾) ↔ ∀𝑘𝐴 if( (𝐹𝑘) ≈ 1𝑜, (𝐹𝑘), (𝑔𝑘)) ∈ ( (𝐹𝑘) ∖ 𝐾)))
171154, 170anbi12d 746 . . . . 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))))
172171spcegv 3285 . . . 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))))
1731723impib 1259 . . 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)))
174116, 125, 153, 173syl3anc 1323 . 2 ((𝜑 ∧ (𝑔:{𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}⟶V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝑔𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
175113, 174exlimddv 1865 1 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  {cab 2612  wral 2912  wrex 2913  {crab 2916  Vcvv 3191  cdif 3557  cin 3559  wss 3560  c0 3896  ifcif 4063  𝒫 cpw 4135  {csn 4153   cuni 4407   ciun 4490   class class class wbr 4618  cmpt 4678  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082   Fn wfn 5845  wf 5846  cfv 5850  cmpt2 6607  1𝑜c1o 7499  Xcixp 7853  cen 7897  Fincfn 7900  cardccrd 8706  Compccmp 21094  UFLcufl 21609
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-omul 7511  df-er 7688  df-map 7805  df-ixp 7854  df-en 7901  df-dom 7902  df-fin 7904  df-wdom 8409  df-card 8710  df-acn 8713  df-cmp 21095
This theorem is referenced by:  ptcmplem4  21764
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