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Theorem hashbclem 13806
Description: Lemma for hashbc 13807: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)
Hypotheses
Ref Expression
hashbc.1 (𝜑𝐴 ∈ Fin)
hashbc.2 (𝜑 → ¬ 𝑧𝐴)
hashbc.3 (𝜑 → ∀𝑗 ∈ ℤ ((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}))
hashbc.4 (𝜑𝐾 ∈ ℤ)
Assertion
Ref Expression
hashbclem (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}))
Distinct variable groups:   𝑥,𝑗,𝑧,𝐴   𝑗,𝐾,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑧,𝑗)   𝐾(𝑧)

Proof of Theorem hashbclem
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7148 . . . . . 6 (𝑗 = 𝐾 → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C𝐾))
2 eqeq2 2834 . . . . . . . 8 (𝑗 = 𝐾 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = 𝐾))
32rabbidv 3455 . . . . . . 7 (𝑗 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})
43fveq2d 6656 . . . . . 6 (𝑗 = 𝐾 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
51, 4eqeq12d 2838 . . . . 5 (𝑗 = 𝐾 → (((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})))
6 hashbc.3 . . . . 5 (𝜑 → ∀𝑗 ∈ ℤ ((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}))
7 hashbc.4 . . . . 5 (𝜑𝐾 ∈ ℤ)
85, 6, 7rspcdva 3600 . . . 4 (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
9 ssun1 4123 . . . . . . . . . . . . 13 𝐴 ⊆ (𝐴 ∪ {𝑧})
109sspwi 4525 . . . . . . . . . . . 12 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ {𝑧})
1110sseli 3938 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
1211adantl 485 . . . . . . . . . 10 ((𝜑𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}))
13 hashbc.2 . . . . . . . . . . 11 (𝜑 → ¬ 𝑧𝐴)
14 elpwi 4520 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1514ssneld 3944 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝐴 → (¬ 𝑧𝐴 → ¬ 𝑧𝑥))
1613, 15mpan9 510 . . . . . . . . . 10 ((𝜑𝑥 ∈ 𝒫 𝐴) → ¬ 𝑧𝑥)
1712, 16jca 515 . . . . . . . . 9 ((𝜑𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥))
18 elpwi 4520 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ (𝐴 ∪ {𝑧}))
19 uncom 4104 . . . . . . . . . . . . . 14 (𝐴 ∪ {𝑧}) = ({𝑧} ∪ 𝐴)
2018, 19sseqtrdi 3992 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
2120adantr 484 . . . . . . . . . . . 12 ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥) → 𝑥 ⊆ ({𝑧} ∪ 𝐴))
22 simpr 488 . . . . . . . . . . . . . 14 ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥) → ¬ 𝑧𝑥)
23 disjsn 4621 . . . . . . . . . . . . . 14 ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑥)
2422, 23sylibr 237 . . . . . . . . . . . . 13 ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥) → (𝑥 ∩ {𝑧}) = ∅)
25 disjssun 4389 . . . . . . . . . . . . 13 ((𝑥 ∩ {𝑧}) = ∅ → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥𝐴))
2624, 25syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥) → (𝑥 ⊆ ({𝑧} ∪ 𝐴) ↔ 𝑥𝐴))
2721, 26mpbid 235 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥) → 𝑥𝐴)
28 vex 3472 . . . . . . . . . . . 12 𝑥 ∈ V
2928elpw 4515 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3027, 29sylibr 237 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥) → 𝑥 ∈ 𝒫 𝐴)
3130adantl 485 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥)) → 𝑥 ∈ 𝒫 𝐴)
3217, 31impbida 800 . . . . . . . 8 (𝜑 → (𝑥 ∈ 𝒫 𝐴 ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥)))
3332anbi1d 632 . . . . . . 7 (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (♯‘𝑥) = 𝐾) ↔ ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥) ∧ (♯‘𝑥) = 𝐾)))
34 anass 472 . . . . . . 7 (((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)))
3533, 34syl6bb 290 . . . . . 6 (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))))
3635rabbidva2 3451 . . . . 5 (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})
3736fveq2d 6656 . . . 4 (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
388, 37eqtrd 2857 . . 3 (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
39 oveq2 7148 . . . . . 6 (𝑗 = (𝐾 − 1) → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C(𝐾 − 1)))
40 eqeq2 2834 . . . . . . . 8 (𝑗 = (𝐾 − 1) → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = (𝐾 − 1)))
4140rabbidv 3455 . . . . . . 7 (𝑗 = (𝐾 − 1) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
4241fveq2d 6656 . . . . . 6 (𝑗 = (𝐾 − 1) → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
4339, 42eqeq12d 2838 . . . . 5 (𝑗 = (𝐾 − 1) → (((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})))
44 peano2zm 12013 . . . . . 6 (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
457, 44syl 17 . . . . 5 (𝜑 → (𝐾 − 1) ∈ ℤ)
4643, 6, 45rspcdva 3600 . . . 4 (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
47 hashbc.1 . . . . . . . 8 (𝜑𝐴 ∈ Fin)
48 pwfi 8807 . . . . . . . 8 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
4947, 48sylib 221 . . . . . . 7 (𝜑 → 𝒫 𝐴 ∈ Fin)
50 rabexg 5210 . . . . . . 7 (𝒫 𝐴 ∈ Fin → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ V)
5149, 50syl 17 . . . . . 6 (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ V)
52 snfi 8581 . . . . . . . . . 10 {𝑧} ∈ Fin
53 unfi 8773 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
5447, 52, 53sylancl 589 . . . . . . . . 9 (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
55 pwfi 8807 . . . . . . . . 9 ((𝐴 ∪ {𝑧}) ∈ Fin ↔ 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
5654, 55sylib 221 . . . . . . . 8 (𝜑 → 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
57 ssrab2 4031 . . . . . . . 8 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
58 ssfi 8726 . . . . . . . 8 ((𝒫 (𝐴 ∪ {𝑧}) ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})) → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
5956, 57, 58sylancl 589 . . . . . . 7 (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
6059elexd 3489 . . . . . 6 (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ V)
61 fveqeq2 6661 . . . . . . . 8 (𝑥 = 𝑢 → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘𝑢) = (𝐾 − 1)))
6261elrab 3655 . . . . . . 7 (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ↔ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)))
63 eleq2 2902 . . . . . . . . . 10 (𝑥 = (𝑢 ∪ {𝑧}) → (𝑧𝑥𝑧 ∈ (𝑢 ∪ {𝑧})))
64 fveqeq2 6661 . . . . . . . . . 10 (𝑥 = (𝑢 ∪ {𝑧}) → ((♯‘𝑥) = 𝐾 ↔ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
6563, 64anbi12d 633 . . . . . . . . 9 (𝑥 = (𝑢 ∪ {𝑧}) → ((𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾)))
66 elpwi 4520 . . . . . . . . . . . 12 (𝑢 ∈ 𝒫 𝐴𝑢𝐴)
6766ad2antrl 727 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢𝐴)
68 unss1 4130 . . . . . . . . . . 11 (𝑢𝐴 → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
6967, 68syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
70 vex 3472 . . . . . . . . . . . 12 𝑢 ∈ V
71 snex 5309 . . . . . . . . . . . 12 {𝑧} ∈ V
7270, 71unex 7454 . . . . . . . . . . 11 (𝑢 ∪ {𝑧}) ∈ V
7372elpw 4515 . . . . . . . . . 10 ((𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
7469, 73sylibr 237 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}))
7547adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐴 ∈ Fin)
7675, 67ssfid 8729 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ∈ Fin)
7752a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → {𝑧} ∈ Fin)
7813adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧𝐴)
7967, 78ssneldd 3945 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧𝑢)
80 disjsn 4621 . . . . . . . . . . . . 13 ((𝑢 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑢)
8179, 80sylibr 237 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∩ {𝑧}) = ∅)
82 hashun 13739 . . . . . . . . . . . 12 ((𝑢 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝑢 ∩ {𝑧}) = ∅) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
8376, 77, 81, 82syl3anc 1368 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
84 simprr 772 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘𝑢) = (𝐾 − 1))
85 hashsng 13726 . . . . . . . . . . . . . 14 (𝑧 ∈ V → (♯‘{𝑧}) = 1)
8685elv 3474 . . . . . . . . . . . . 13 (♯‘{𝑧}) = 1
8786a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘{𝑧}) = 1)
8884, 87oveq12d 7158 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ((♯‘𝑢) + (♯‘{𝑧})) = ((𝐾 − 1) + 1))
897adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℤ)
9089zcnd 12076 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℂ)
91 ax-1cn 10584 . . . . . . . . . . . 12 1 ∈ ℂ
92 npcan 10884 . . . . . . . . . . . 12 ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
9390, 91, 92sylancl 589 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ((𝐾 − 1) + 1) = 𝐾)
9483, 88, 933eqtrd 2861 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = 𝐾)
95 ssun2 4124 . . . . . . . . . . 11 {𝑧} ⊆ (𝑢 ∪ {𝑧})
96 vex 3472 . . . . . . . . . . . 12 𝑧 ∈ V
9796snss 4692 . . . . . . . . . . 11 (𝑧 ∈ (𝑢 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝑢 ∪ {𝑧}))
9895, 97mpbir 234 . . . . . . . . . 10 𝑧 ∈ (𝑢 ∪ {𝑧})
9994, 98jctil 523 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
10065, 74, 99elrabd 3657 . . . . . . . 8 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})
101100ex 416 . . . . . . 7 (𝜑 → ((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
10262, 101syl5bi 245 . . . . . 6 (𝜑 → (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
103 eleq2 2902 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑧𝑥𝑧𝑣))
104 fveqeq2 6661 . . . . . . . . 9 (𝑥 = 𝑣 → ((♯‘𝑥) = 𝐾 ↔ (♯‘𝑣) = 𝐾))
105103, 104anbi12d 633 . . . . . . . 8 (𝑥 = 𝑣 → ((𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾)))
106105elrab 3655 . . . . . . 7 (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ↔ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾)))
107 fveqeq2 6661 . . . . . . . . 9 (𝑥 = (𝑣 ∖ {𝑧}) → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1)))
108 elpwi 4520 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
109108ad2antrl 727 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
110109, 19sseqtrdi 3992 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ ({𝑧} ∪ 𝐴))
111 ssundif 4405 . . . . . . . . . . 11 (𝑣 ⊆ ({𝑧} ∪ 𝐴) ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
112110, 111sylib 221 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ⊆ 𝐴)
113 vex 3472 . . . . . . . . . . . 12 𝑣 ∈ V
114113difexi 5208 . . . . . . . . . . 11 (𝑣 ∖ {𝑧}) ∈ V
115114elpw 4515 . . . . . . . . . 10 ((𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴 ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
116112, 115sylibr 237 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴)
11747adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝐴 ∈ Fin)
118117, 112ssfid 8729 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ Fin)
119 hashcl 13713 . . . . . . . . . . . . 13 ((𝑣 ∖ {𝑧}) ∈ Fin → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ0)
120118, 119syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ0)
121120nn0cnd 11945 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℂ)
122 pncan 10881 . . . . . . . . . . 11 (((♯‘(𝑣 ∖ {𝑧})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
123121, 91, 122sylancl 589 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
124 undif1 4396 . . . . . . . . . . . . . 14 ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = (𝑣 ∪ {𝑧})
125 simprrl 780 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧𝑣)
126125snssd 4715 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
127 ssequn2 4134 . . . . . . . . . . . . . . 15 ({𝑧} ⊆ 𝑣 ↔ (𝑣 ∪ {𝑧}) = 𝑣)
128126, 127sylib 221 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∪ {𝑧}) = 𝑣)
129124, 128syl5eq 2869 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = 𝑣)
130129fveq2d 6656 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = (♯‘𝑣))
13152a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ∈ Fin)
132 incom 4152 . . . . . . . . . . . . . . . 16 ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ({𝑧} ∩ (𝑣 ∖ {𝑧}))
133 disjdif 4393 . . . . . . . . . . . . . . . 16 ({𝑧} ∩ (𝑣 ∖ {𝑧})) = ∅
134132, 133eqtri 2845 . . . . . . . . . . . . . . 15 ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅
135134a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅)
136 hashun 13739 . . . . . . . . . . . . . 14 (((𝑣 ∖ {𝑧}) ∈ Fin ∧ {𝑧} ∈ Fin ∧ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
137118, 131, 135, 136syl3anc 1368 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
13886oveq2i 7151 . . . . . . . . . . . . 13 ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1)
139137, 138syl6eq 2873 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1))
140 simprrr 781 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘𝑣) = 𝐾)
141130, 139, 1403eqtr3d 2865 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((♯‘(𝑣 ∖ {𝑧})) + 1) = 𝐾)
142141oveq1d 7155 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (𝐾 − 1))
143123, 142eqtr3d 2859 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1))
144107, 116, 143elrabd 3657 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
145144ex 416 . . . . . . 7 (𝜑 → ((𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾)) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
146106, 145syl5bi 245 . . . . . 6 (𝜑 → (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
14762, 106anbi12i 629 . . . . . . 7 ((𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) ↔ ((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))))
148 simp3rl 1243 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧𝑣)
149148snssd 4715 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
150 incom 4152 . . . . . . . . . . . 12 ({𝑧} ∩ 𝑢) = (𝑢 ∩ {𝑧})
151813adant3 1129 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 ∩ {𝑧}) = ∅)
152150, 151syl5eq 2869 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ({𝑧} ∩ 𝑢) = ∅)
153 uneqdifeq 4410 . . . . . . . . . . 11 (({𝑧} ⊆ 𝑣 ∧ ({𝑧} ∩ 𝑢) = ∅) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢))
154149, 152, 153syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢))
155154bicomd 226 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) = 𝑢 ↔ ({𝑧} ∪ 𝑢) = 𝑣))
156 eqcom 2829 . . . . . . . . 9 (𝑢 = (𝑣 ∖ {𝑧}) ↔ (𝑣 ∖ {𝑧}) = 𝑢)
157 eqcom 2829 . . . . . . . . . 10 (𝑣 = (𝑢 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) = 𝑣)
158 uncom 4104 . . . . . . . . . . 11 (𝑢 ∪ {𝑧}) = ({𝑧} ∪ 𝑢)
159158eqeq1i 2827 . . . . . . . . . 10 ((𝑢 ∪ {𝑧}) = 𝑣 ↔ ({𝑧} ∪ 𝑢) = 𝑣)
160157, 159bitri 278 . . . . . . . . 9 (𝑣 = (𝑢 ∪ {𝑧}) ↔ ({𝑧} ∪ 𝑢) = 𝑣)
161155, 156, 1603bitr4g 317 . . . . . . . 8 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧})))
1621613expib 1119 . . . . . . 7 (𝜑 → (((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
163147, 162syl5bi 245 . . . . . 6 (𝜑 → ((𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
16451, 60, 102, 146, 163en3d 8533 . . . . 5 (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})
165 ssrab2 4031 . . . . . . 7 {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴
166 ssfi 8726 . . . . . . 7 ((𝒫 𝐴 ∈ Fin ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin)
16749, 165, 166sylancl 589 . . . . . 6 (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin)
168 hashen 13703 . . . . . 6 (({𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin) → ((♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) ↔ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
169167, 59, 168syl2anc 587 . . . . 5 (𝜑 → ((♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) ↔ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
170164, 169mpbird 260 . . . 4 (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
17146, 170eqtrd 2857 . . 3 (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
17238, 171oveq12d 7158 . 2 (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})))
17352a1i 11 . . . . . 6 (𝜑 → {𝑧} ∈ Fin)
174 disjsn 4621 . . . . . . 7 ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝐴)
17513, 174sylibr 237 . . . . . 6 (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
176 hashun 13739 . . . . . 6 ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝐴 ∩ {𝑧}) = ∅) → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
17747, 173, 175, 176syl3anc 1368 . . . . 5 (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
17886oveq2i 7151 . . . . 5 ((♯‘𝐴) + (♯‘{𝑧})) = ((♯‘𝐴) + 1)
179177, 178syl6eq 2873 . . . 4 (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + 1))
180179oveq1d 7155 . . 3 (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴) + 1)C𝐾))
181 hashcl 13713 . . . . 5 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
18247, 181syl 17 . . . 4 (𝜑 → (♯‘𝐴) ∈ ℕ0)
183 bcpasc 13677 . . . 4 (((♯‘𝐴) ∈ ℕ0𝐾 ∈ ℤ) → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
184182, 7, 183syl2anc 587 . . 3 (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
185180, 184eqtr4d 2860 . 2 (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))))
186 pm2.1 894 . . . . . . . 8 𝑧𝑥𝑧𝑥)
187186biantrur 534 . . . . . . 7 ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧𝑥𝑧𝑥) ∧ (♯‘𝑥) = 𝐾))
188 andir 1006 . . . . . . 7 (((¬ 𝑧𝑥𝑧𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)))
189187, 188bitri 278 . . . . . 6 ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)))
190189rabbii 3448 . . . . 5 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))}
191 unrab 4248 . . . . 5 ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))}
192190, 191eqtr4i 2848 . . . 4 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾} = ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})
193192fveq2i 6655 . . 3 (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}) = (♯‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
194 ssrab2 4031 . . . . 5 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
195 ssfi 8726 . . . . 5 ((𝒫 (𝐴 ∪ {𝑧}) ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})) → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
19656, 194, 195sylancl 589 . . . 4 (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
197 inrab 4249 . . . . . 6 ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))}
198 simprl 770 . . . . . . . . 9 (((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)) → 𝑧𝑥)
199 simpll 766 . . . . . . . . 9 (((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)) → ¬ 𝑧𝑥)
200198, 199pm2.65i 197 . . . . . . . 8 ¬ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))
201200rgenw 3142 . . . . . . 7 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))
202 rabeq0 4310 . . . . . . 7 ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅ ↔ ∀𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)))
203201, 202mpbir 234 . . . . . 6 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅
204197, 203eqtri 2845 . . . . 5 ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅
205204a1i 11 . . . 4 (𝜑 → ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅)
206 hashun 13739 . . . 4 (({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin ∧ ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅) → (♯‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})))
207196, 59, 205, 206syl3anc 1368 . . 3 (𝜑 → (♯‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})))
208193, 207syl5eq 2869 . 2 (𝜑 → (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})))
209172, 185, 2083eqtr4d 2867 1 (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2114  wral 3130  {crab 3134  Vcvv 3469  cdif 3905  cun 3906  cin 3907  wss 3908  c0 4265  𝒫 cpw 4511  {csn 4539   class class class wbr 5042  cfv 6334  (class class class)co 7140  cen 8493  Fincfn 8496  cc 10524  1c1 10527   + caddc 10529  cmin 10859  0cn0 11885  cz 11969  Ccbc 13658  chash 13686
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-seq 13365  df-fac 13630  df-bc 13659  df-hash 13687
This theorem is referenced by:  hashbc  13807
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