Theorem ballotfilemelo 13141

Description: Elementhood in 𝑂. (Contributed by Thierry Arnoux, 17-Apr-2017.)

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotfi.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}

Assertion

Ref	Expression
ballotfilemelo	⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ Fin ∧ (♯‘𝐶) = 𝑀))

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐

Allowed substitution hint:   𝐶(𝑐)

Proof of Theorem ballotfilemelo

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfpw 7215	. . 3 ⊢ (𝐶 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ Fin))
2	1	anbi1i 458	. 2 ⊢ ((𝐶 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝐶) = 𝑀) ↔ ((𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ Fin) ∧ (♯‘𝐶) = 𝑀))
3		fveqeq2 5679	. . 3 ⊢ (𝑑 = 𝐶 → ((♯‘𝑑) = 𝑀 ↔ (♯‘𝐶) = 𝑀))
4		ballotfi.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
5		fveqeq2 5679	. . . . 5 ⊢ (𝑐 = 𝑑 → ((♯‘𝑐) = 𝑀 ↔ (♯‘𝑑) = 𝑀))
6	5	cbvrabv 2812	. . . 4 ⊢ {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀} = {𝑑 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑑) = 𝑀}
7	4, 6	eqtri 2253	. . 3 ⊢ 𝑂 = {𝑑 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑑) = 𝑀}
8	3, 7	elrab2 2976	. 2 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝐶) = 𝑀))
9		df-3an 1007	. 2 ⊢ ((𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ Fin ∧ (♯‘𝐶) = 𝑀) ↔ ((𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ Fin) ∧ (♯‘𝐶) = 𝑀))
10	2, 8, 9	3bitr4i 212	1 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ Fin ∧ (♯‘𝐶) = 𝑀))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  {crab 2524   ∩ cin 3210   ⊆ wss 3211  𝒫 cpw 3669  ‘cfv 5352  (class class class)co 6050  Fincfn 6975  1c1 8128   + caddc 8130  ℕcn 9237  ...cfz 10342  ♯chash 11138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360

This theorem is referenced by: (None)