Theorem ballotfilemelo 13200

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 7252	. . 3 ⊢ (𝐶 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ Fin))
2	1	anbi1i 462	. 2 ⊢ ((𝐶 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝐶) = 𝑀) ↔ ((𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝐶 ∈ Fin) ∧ (♯‘𝐶) = 𝑀))
3		fveqeq2 5699	. . 3 ⊢ (𝑑 = 𝐶 → ((♯‘𝑑) = 𝑀 ↔ (♯‘𝐶) = 𝑀))
4		ballotfilem.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
5		fveqeq2 5699	. . . . 5 ⊢ (𝑐 = 𝑑 → ((♯‘𝑐) = 𝑀 ↔ (♯‘𝑑) = 𝑀))
6	5	cbvrabv 2820	. . . 4 ⊢ {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀} = {𝑑 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑑) = 𝑀}
7	4, 6	eqtri 2259	. . 3 ⊢ 𝑂 = {𝑑 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑑) = 𝑀}
8	3, 7	elrab2 2985	. 2 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝐶) = 𝑀))
9		df-3an 1011	. 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 1009   = wceq 1402   ∈ wcel 2209  {crab 2532   ∩ cin 3219   ⊆ wss 3220  𝒫 cpw 3685  ‘cfv 5372  (class class class)co 6075  Fincfn 7012  1c1 8170   + caddc 8172  ℕcn 9283  ...cfz 10390  ♯chash 11192

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380

This theorem is referenced by:  ballotfilemcdc  13201  ballotfilemfc0  13210  ballotfilemscr  13240  ballotfilemro  13244  ballotfilemrinv0  13254