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Theorem hashbclem 14029
Description: Lemma for hashbc 14030: 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 7230 . . . . . 6 (𝑗 = 𝐾 → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C𝐾))
2 eqeq2 2750 . . . . . . . 8 (𝑗 = 𝐾 → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = 𝐾))
32rabbidv 3397 . . . . . . 7 (𝑗 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾})
43fveq2d 6730 . . . . . 6 (𝑗 = 𝐾 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
51, 4eqeq12d 2754 . . . . 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 3546 . . . 4 (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}))
9 ssun1 4095 . . . . . . . . . . . . 13 𝐴 ⊆ (𝐴 ∪ {𝑧})
109sspwi 4536 . . . . . . . . . . . 12 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ {𝑧})
1110sseli 3905 . . . . . . . . . . 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 4531 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1514ssneld 3912 . . . . . . . . . . 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 4531 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑥 ⊆ (𝐴 ∪ {𝑧}))
19 uncom 4076 . . . . . . . . . . . . . 14 (𝐴 ∪ {𝑧}) = ({𝑧} ∪ 𝐴)
2018, 19sseqtrdi 3960 . . . . . . . . . . . . 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 4636 . . . . . . . . . . . . . 14 ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑥)
2422, 23sylibr 237 . . . . . . . . . . . . 13 ((𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥) → (𝑥 ∩ {𝑧}) = ∅)
25 disjssun 4391 . . . . . . . . . . . . 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 3419 . . . . . . . . . . . 12 𝑥 ∈ V
2928elpw 4526 . . . . . . . . . . 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 801 . . . . . . . 8 (𝜑 → (𝑥 ∈ 𝒫 𝐴 ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ ¬ 𝑧𝑥)))
3332anbi1d 633 . . . . . . 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, 34bitrdi 290 . . . . . 6 (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))))
3635rabbidva2 3393 . . . . 5 (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})
3736fveq2d 6730 . . . 4 (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝐾}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
388, 37eqtrd 2778 . . 3 (𝜑 → ((♯‘𝐴)C𝐾) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
39 oveq2 7230 . . . . . 6 (𝑗 = (𝐾 − 1) → ((♯‘𝐴)C𝑗) = ((♯‘𝐴)C(𝐾 − 1)))
40 eqeq2 2750 . . . . . . . 8 (𝑗 = (𝐾 − 1) → ((♯‘𝑥) = 𝑗 ↔ (♯‘𝑥) = (𝐾 − 1)))
4140rabbidv 3397 . . . . . . 7 (𝑗 = (𝐾 − 1) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
4241fveq2d 6730 . . . . . 6 (𝑗 = (𝐾 − 1) → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
4339, 42eqeq12d 2754 . . . . 5 (𝑗 = (𝐾 − 1) → (((♯‘𝐴)C𝑗) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑗}) ↔ ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})))
44 peano2zm 12233 . . . . . 6 (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
457, 44syl 17 . . . . 5 (𝜑 → (𝐾 − 1) ∈ ℤ)
4643, 6, 45rspcdva 3546 . . . 4 (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
47 hashbc.1 . . . . . . . 8 (𝜑𝐴 ∈ Fin)
48 pwfi 8867 . . . . . . . 8 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
4947, 48sylib 221 . . . . . . 7 (𝜑 → 𝒫 𝐴 ∈ Fin)
50 rabexg 5233 . . . . . . 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 8732 . . . . . . . . 9 {𝑧} ∈ Fin
53 unfi 8861 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
5447, 52, 53sylancl 589 . . . . . . . 8 (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
55 pwfi 8867 . . . . . . . 8 ((𝐴 ∪ {𝑧}) ∈ Fin ↔ 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
5654, 55sylib 221 . . . . . . 7 (𝜑 → 𝒫 (𝐴 ∪ {𝑧}) ∈ Fin)
57 ssrab2 4002 . . . . . . 7 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
58 ssfi 8862 . . . . . . 7 ((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 . . . . . 6 (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
60 fveqeq2 6735 . . . . . . . 8 (𝑥 = 𝑢 → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘𝑢) = (𝐾 − 1)))
6160elrab 3609 . . . . . . 7 (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ↔ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)))
62 eleq2 2827 . . . . . . . . . 10 (𝑥 = (𝑢 ∪ {𝑧}) → (𝑧𝑥𝑧 ∈ (𝑢 ∪ {𝑧})))
63 fveqeq2 6735 . . . . . . . . . 10 (𝑥 = (𝑢 ∪ {𝑧}) → ((♯‘𝑥) = 𝐾 ↔ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
6462, 63anbi12d 634 . . . . . . . . 9 (𝑥 = (𝑢 ∪ {𝑧}) → ((𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾)))
65 elpwi 4531 . . . . . . . . . . . 12 (𝑢 ∈ 𝒫 𝐴𝑢𝐴)
6665ad2antrl 728 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢𝐴)
67 unss1 4102 . . . . . . . . . . 11 (𝑢𝐴 → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
6866, 67syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
69 vex 3419 . . . . . . . . . . . 12 𝑢 ∈ V
70 snex 5333 . . . . . . . . . . . 12 {𝑧} ∈ V
7169, 70unex 7540 . . . . . . . . . . 11 (𝑢 ∪ {𝑧}) ∈ V
7271elpw 4526 . . . . . . . . . 10 ((𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) ⊆ (𝐴 ∪ {𝑧}))
7368, 72sylibr 237 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ 𝒫 (𝐴 ∪ {𝑧}))
7447adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐴 ∈ Fin)
7574, 66ssfid 8911 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝑢 ∈ Fin)
7652a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → {𝑧} ∈ Fin)
7713adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧𝐴)
7866, 77ssneldd 3913 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ¬ 𝑧𝑢)
79 disjsn 4636 . . . . . . . . . . . . 13 ((𝑢 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑢)
8078, 79sylibr 237 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∩ {𝑧}) = ∅)
81 hashun 13962 . . . . . . . . . . . 12 ((𝑢 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝑢 ∩ {𝑧}) = ∅) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
8275, 76, 80, 81syl3anc 1373 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = ((♯‘𝑢) + (♯‘{𝑧})))
83 simprr 773 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘𝑢) = (𝐾 − 1))
84 hashsng 13949 . . . . . . . . . . . . . 14 (𝑧 ∈ V → (♯‘{𝑧}) = 1)
8584elv 3421 . . . . . . . . . . . . 13 (♯‘{𝑧}) = 1
8685a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘{𝑧}) = 1)
8783, 86oveq12d 7240 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ((♯‘𝑢) + (♯‘{𝑧})) = ((𝐾 − 1) + 1))
887adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℤ)
8988zcnd 12296 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → 𝐾 ∈ ℂ)
90 ax-1cn 10800 . . . . . . . . . . . 12 1 ∈ ℂ
91 npcan 11100 . . . . . . . . . . . 12 ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
9289, 90, 91sylancl 589 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → ((𝐾 − 1) + 1) = 𝐾)
9382, 87, 923eqtrd 2782 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (♯‘(𝑢 ∪ {𝑧})) = 𝐾)
94 ssun2 4096 . . . . . . . . . . 11 {𝑧} ⊆ (𝑢 ∪ {𝑧})
95 vex 3419 . . . . . . . . . . . 12 𝑧 ∈ V
9695snss 4708 . . . . . . . . . . 11 (𝑧 ∈ (𝑢 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝑢 ∪ {𝑧}))
9794, 96mpbir 234 . . . . . . . . . 10 𝑧 ∈ (𝑢 ∪ {𝑧})
9893, 97jctil 523 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑧 ∈ (𝑢 ∪ {𝑧}) ∧ (♯‘(𝑢 ∪ {𝑧})) = 𝐾))
9964, 73, 98elrabd 3611 . . . . . . . 8 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1))) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})
10099ex 416 . . . . . . 7 (𝜑 → ((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
10161, 100syl5bi 245 . . . . . 6 (𝜑 → (𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} → (𝑢 ∪ {𝑧}) ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
102 eleq2 2827 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑧𝑥𝑧𝑣))
103 fveqeq2 6735 . . . . . . . . 9 (𝑥 = 𝑣 → ((♯‘𝑥) = 𝐾 ↔ (♯‘𝑣) = 𝐾))
104102, 103anbi12d 634 . . . . . . . 8 (𝑥 = 𝑣 → ((𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ↔ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾)))
105104elrab 3609 . . . . . . 7 (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ↔ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾)))
106 fveqeq2 6735 . . . . . . . . 9 (𝑥 = (𝑣 ∖ {𝑧}) → ((♯‘𝑥) = (𝐾 − 1) ↔ (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1)))
107 elpwi 4531 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
108107ad2antrl 728 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ (𝐴 ∪ {𝑧}))
109108, 19sseqtrdi 3960 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑣 ⊆ ({𝑧} ∪ 𝐴))
110 ssundif 4408 . . . . . . . . . . 11 (𝑣 ⊆ ({𝑧} ∪ 𝐴) ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
111109, 110sylib 221 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ⊆ 𝐴)
112 vex 3419 . . . . . . . . . . . 12 𝑣 ∈ V
113112difexi 5230 . . . . . . . . . . 11 (𝑣 ∖ {𝑧}) ∈ V
114113elpw 4526 . . . . . . . . . 10 ((𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴 ↔ (𝑣 ∖ {𝑧}) ⊆ 𝐴)
115111, 114sylibr 237 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ 𝒫 𝐴)
11647adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝐴 ∈ Fin)
117116, 111ssfid 8911 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ Fin)
118 hashcl 13936 . . . . . . . . . . . . 13 ((𝑣 ∖ {𝑧}) ∈ Fin → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ0)
119117, 118syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℕ0)
120119nn0cnd 12165 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) ∈ ℂ)
121 pncan 11097 . . . . . . . . . . 11 (((♯‘(𝑣 ∖ {𝑧})) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
122120, 90, 121sylancl 589 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (♯‘(𝑣 ∖ {𝑧})))
123 undif1 4399 . . . . . . . . . . . . . 14 ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = (𝑣 ∪ {𝑧})
124 simprrl 781 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧𝑣)
125124snssd 4731 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
126 ssequn2 4106 . . . . . . . . . . . . . . 15 ({𝑧} ⊆ 𝑣 ↔ (𝑣 ∪ {𝑧}) = 𝑣)
127125, 126sylib 221 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∪ {𝑧}) = 𝑣)
128123, 127eqtrid 2790 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∪ {𝑧}) = 𝑣)
129128fveq2d 6730 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = (♯‘𝑣))
13052a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ∈ Fin)
131 disjdifr 4396 . . . . . . . . . . . . . . 15 ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅
132131a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅)
133 hashun 13962 . . . . . . . . . . . . . 14 (((𝑣 ∖ {𝑧}) ∈ Fin ∧ {𝑧} ∈ Fin ∧ ((𝑣 ∖ {𝑧}) ∩ {𝑧}) = ∅) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
134117, 130, 132, 133syl3anc 1373 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})))
13585oveq2i 7233 . . . . . . . . . . . . 13 ((♯‘(𝑣 ∖ {𝑧})) + (♯‘{𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1)
136134, 135eqtrdi 2795 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘((𝑣 ∖ {𝑧}) ∪ {𝑧})) = ((♯‘(𝑣 ∖ {𝑧})) + 1))
137 simprrr 782 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘𝑣) = 𝐾)
138129, 136, 1373eqtr3d 2786 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((♯‘(𝑣 ∖ {𝑧})) + 1) = 𝐾)
139138oveq1d 7237 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (((♯‘(𝑣 ∖ {𝑧})) + 1) − 1) = (𝐾 − 1))
140122, 139eqtr3d 2780 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (♯‘(𝑣 ∖ {𝑧})) = (𝐾 − 1))
141106, 115, 140elrabd 3611 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)})
142141ex 416 . . . . . . 7 (𝜑 → ((𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾)) → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
143105, 142syl5bi 245 . . . . . 6 (𝜑 → (𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} → (𝑣 ∖ {𝑧}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}))
14461, 105anbi12i 630 . . . . . . 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))))
145 simp3rl 1248 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → 𝑧𝑣)
146145snssd 4731 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → {𝑧} ⊆ 𝑣)
147 incom 4124 . . . . . . . . . . . 12 ({𝑧} ∩ 𝑢) = (𝑢 ∩ {𝑧})
148803adant3 1134 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 ∩ {𝑧}) = ∅)
149147, 148eqtrid 2790 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ({𝑧} ∩ 𝑢) = ∅)
150 uneqdifeq 4413 . . . . . . . . . . 11 (({𝑧} ⊆ 𝑣 ∧ ({𝑧} ∩ 𝑢) = ∅) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢))
151146, 149, 150syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (({𝑧} ∪ 𝑢) = 𝑣 ↔ (𝑣 ∖ {𝑧}) = 𝑢))
152151bicomd 226 . . . . . . . . 9 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → ((𝑣 ∖ {𝑧}) = 𝑢 ↔ ({𝑧} ∪ 𝑢) = 𝑣))
153 eqcom 2745 . . . . . . . . 9 (𝑢 = (𝑣 ∖ {𝑧}) ↔ (𝑣 ∖ {𝑧}) = 𝑢)
154 eqcom 2745 . . . . . . . . . 10 (𝑣 = (𝑢 ∪ {𝑧}) ↔ (𝑢 ∪ {𝑧}) = 𝑣)
155 uncom 4076 . . . . . . . . . . 11 (𝑢 ∪ {𝑧}) = ({𝑧} ∪ 𝑢)
156155eqeq1i 2743 . . . . . . . . . 10 ((𝑢 ∪ {𝑧}) = 𝑣 ↔ ({𝑧} ∪ 𝑢) = 𝑣)
157154, 156bitri 278 . . . . . . . . 9 (𝑣 = (𝑢 ∪ {𝑧}) ↔ ({𝑧} ∪ 𝑢) = 𝑣)
158152, 153, 1573bitr4g 317 . . . . . . . 8 ((𝜑 ∧ (𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧})))
1591583expib 1124 . . . . . . 7 (𝜑 → (((𝑢 ∈ 𝒫 𝐴 ∧ (♯‘𝑢) = (𝐾 − 1)) ∧ (𝑣 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∧ (𝑧𝑣 ∧ (♯‘𝑣) = 𝐾))) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
160144, 159syl5bi 245 . . . . . 6 (𝜑 → ((𝑢 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∧ 𝑣 ∈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) → (𝑢 = (𝑣 ∖ {𝑧}) ↔ 𝑣 = (𝑢 ∪ {𝑧}))))
16151, 59, 101, 143, 160en3d 8676 . . . . 5 (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ≈ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})
162 ssrab2 4002 . . . . . . 7 {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴
163 ssfi 8862 . . . . . . 7 ((𝒫 𝐴 ∈ Fin ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ⊆ 𝒫 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin)
16449, 162, 163sylancl 589 . . . . . 6 (𝜑 → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)} ∈ Fin)
165 hashen 13926 . . . . . 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)}))
166164, 59, 165syl2anc 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)}))
167161, 166mpbird 260 . . . 4 (𝜑 → (♯‘{𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = (𝐾 − 1)}) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
16846, 167eqtrd 2778 . . 3 (𝜑 → ((♯‘𝐴)C(𝐾 − 1)) = (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
16938, 168oveq12d 7240 . 2 (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})))
17052a1i 11 . . . . . 6 (𝜑 → {𝑧} ∈ Fin)
171 disjsn 4636 . . . . . . 7 ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝐴)
17213, 171sylibr 237 . . . . . 6 (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
173 hashun 13962 . . . . . 6 ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin ∧ (𝐴 ∩ {𝑧}) = ∅) → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
17447, 170, 172, 173syl3anc 1373 . . . . 5 (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + (♯‘{𝑧})))
17585oveq2i 7233 . . . . 5 ((♯‘𝐴) + (♯‘{𝑧})) = ((♯‘𝐴) + 1)
176174, 175eqtrdi 2795 . . . 4 (𝜑 → (♯‘(𝐴 ∪ {𝑧})) = ((♯‘𝐴) + 1))
177176oveq1d 7237 . . 3 (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴) + 1)C𝐾))
178 hashcl 13936 . . . . 5 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
17947, 178syl 17 . . . 4 (𝜑 → (♯‘𝐴) ∈ ℕ0)
180 bcpasc 13900 . . . 4 (((♯‘𝐴) ∈ ℕ0𝐾 ∈ ℤ) → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
181179, 7, 180syl2anc 587 . . 3 (𝜑 → (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))) = (((♯‘𝐴) + 1)C𝐾))
182177, 181eqtr4d 2781 . 2 (𝜑 → ((♯‘(𝐴 ∪ {𝑧}))C𝐾) = (((♯‘𝐴)C𝐾) + ((♯‘𝐴)C(𝐾 − 1))))
183 pm2.1 897 . . . . . . . 8 𝑧𝑥𝑧𝑥)
184183biantrur 534 . . . . . . 7 ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧𝑥𝑧𝑥) ∧ (♯‘𝑥) = 𝐾))
185 andir 1009 . . . . . . 7 (((¬ 𝑧𝑥𝑧𝑥) ∧ (♯‘𝑥) = 𝐾) ↔ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)))
186184, 185bitri 278 . . . . . 6 ((♯‘𝑥) = 𝐾 ↔ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)))
187186rabbii 3390 . . . . 5 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾} = {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∨ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))}
188 unrab 4229 . . . . 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))}
189187, 188eqtr4i 2769 . . . 4 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾} = ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})
190189fveq2i 6729 . . 3 (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}) = (♯‘({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∪ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}))
191 ssrab2 4002 . . . . 5 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})
192 ssfi 8862 . . . . 5 ((𝒫 (𝐴 ∪ {𝑧}) ∈ Fin ∧ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ⊆ 𝒫 (𝐴 ∪ {𝑧})) → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
19356, 191, 192sylancl 589 . . . 4 (𝜑 → {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∈ Fin)
194 inrab 4230 . . . . . 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))}
195 simprl 771 . . . . . . . . 9 (((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)) → 𝑧𝑥)
196 simpll 767 . . . . . . . . 9 (((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)) → ¬ 𝑧𝑥)
197195, 196pm2.65i 197 . . . . . . . 8 ¬ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))
198197rgenw 3074 . . . . . . 7 𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))
199 rabeq0 4308 . . . . . . 7 ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅ ↔ ∀𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ¬ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)))
200198, 199mpbir 234 . . . . . 6 {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ ((¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾) ∧ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾))} = ∅
201194, 200eqtri 2766 . . . . 5 ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅
202201a1i 11 . . . 4 (𝜑 → ({𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)} ∩ {𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) = ∅)
203 hashun 13962 . . . 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)})))
204193, 59, 202, 203syl3anc 1373 . . 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)})))
205190, 204eqtrid 2790 . 2 (𝜑 → (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (♯‘𝑥) = 𝐾}) = ((♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (¬ 𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)}) + (♯‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (𝑧𝑥 ∧ (♯‘𝑥) = 𝐾)})))
206169, 182, 2053eqtr4d 2788 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 847  w3a 1089   = wceq 1543  wcel 2111  wral 3062  {crab 3066  Vcvv 3415  cdif 3872  cun 3873  cin 3874  wss 3875  c0 4246  𝒫 cpw 4522  {csn 4550   class class class wbr 5062  cfv 6389  (class class class)co 7222  cen 8632  Fincfn 8635  cc 10740  1c1 10743   + caddc 10745  cmin 11075  0cn0 12103  cz 12189  Ccbc 13881  chash 13909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pow 5267  ax-pr 5331  ax-un 7532  ax-cnex 10798  ax-resscn 10799  ax-1cn 10800  ax-icn 10801  ax-addcl 10802  ax-addrcl 10803  ax-mulcl 10804  ax-mulrcl 10805  ax-mulcom 10806  ax-addass 10807  ax-mulass 10808  ax-distr 10809  ax-i2m1 10810  ax-1ne0 10811  ax-1rid 10812  ax-rnegex 10813  ax-rrecex 10814  ax-cnre 10815  ax-pre-lttri 10816  ax-pre-lttrn 10817  ax-pre-ltadd 10818  ax-pre-mulgt0 10819
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-pss 3894  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4829  df-int 4869  df-iun 4915  df-br 5063  df-opab 5125  df-mpt 5145  df-tr 5171  df-id 5464  df-eprel 5469  df-po 5477  df-so 5478  df-fr 5518  df-we 5520  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-pred 6169  df-ord 6225  df-on 6226  df-lim 6227  df-suc 6228  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-f1 6394  df-fo 6395  df-f1o 6396  df-fv 6397  df-riota 7179  df-ov 7225  df-oprab 7226  df-mpo 7227  df-om 7654  df-1st 7770  df-2nd 7771  df-wrecs 8056  df-recs 8117  df-rdg 8155  df-1o 8211  df-oadd 8215  df-er 8400  df-en 8636  df-dom 8637  df-sdom 8638  df-fin 8639  df-dju 9530  df-card 9568  df-pnf 10882  df-mnf 10883  df-xr 10884  df-ltxr 10885  df-le 10886  df-sub 11077  df-neg 11078  df-div 11503  df-nn 11844  df-n0 12104  df-z 12190  df-uz 12452  df-rp 12600  df-fz 13109  df-seq 13588  df-fac 13853  df-bc 13882  df-hash 13910
This theorem is referenced by:  hashbc  14030
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