ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inimasn GIF version

Theorem inimasn 4803
Description: The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
inimasn (𝐶𝑉 → ((𝐴𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))

Proof of Theorem inimasn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3167 . . 3 (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})))
2 elin 3167 . . . . 5 (⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
32a1i 9 . . . 4 (𝐶𝑉 → (⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
4 vex 2615 . . . . 5 𝑥 ∈ V
5 elimasng 4755 . . . . 5 ((𝐶𝑉𝑥 ∈ V) → (𝑥 ∈ ((𝐴𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵)))
64, 5mpan2 416 . . . 4 (𝐶𝑉 → (𝑥 ∈ ((𝐴𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵)))
7 elimasng 4755 . . . . . 6 ((𝐶𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
84, 7mpan2 416 . . . . 5 (𝐶𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
9 elimasng 4755 . . . . . 6 ((𝐶𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
104, 9mpan2 416 . . . . 5 (𝐶𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
118, 10anbi12d 457 . . . 4 (𝐶𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
123, 6, 113bitr4rd 219 . . 3 (𝐶𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ 𝑥 ∈ ((𝐴𝐵) “ {𝐶})))
131, 12syl5rbb 191 . 2 (𝐶𝑉 → (𝑥 ∈ ((𝐴𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))))
1413eqrdv 2081 1 (𝐶𝑉 → ((𝐴𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  Vcvv 2612  cin 2983  {csn 3422  cop 3425  cima 4404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4407  df-cnv 4409  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator