Theorem ballotfilemelo 13166

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

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotfilem.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 7228	. . 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 5684	. . 3 ⊢ (𝑑 = 𝐶 → ((♯‘𝑑) = 𝑀 ↔ (♯‘𝐶) = 𝑀))
4		ballotfilem.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
5		fveqeq2 5684	. . . . 5 ⊢ (𝑐 = 𝑑 → ((♯‘𝑐) = 𝑀 ↔ (♯‘𝑑) = 𝑀))
6	5	cbvrabv 2814	. . . 4 ⊢ {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀} = {𝑑 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑑) = 𝑀}
7	4, 6	eqtri 2255	. . 3 ⊢ 𝑂 = {𝑑 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑑) = 𝑀}
8	3, 7	elrab2 2979	. 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 2205  {crab 2526   ∩ cin 3213   ⊆ wss 3214  𝒫 cpw 3674  ‘cfv 5357  (class class class)co 6058  Fincfn 6988  1c1 8144   + caddc 8146  ℕcn 9254  ...cfz 10361  ♯chash 11163

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365

This theorem is referenced by:  ballotfilemcdc  13167  ballotfilemfc0  13176  ballotfilemscr  13206  ballotfilemro  13210  ballotfilemrinv0  13220