MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inimasn Structured version   Visualization version   GIF version

Theorem inimasn 5709
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 3940 . . 3 (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})))
2 elin 3940 . . . . 5 (⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
32a1i 11 . . . 4 (𝐶𝑉 → (⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
4 vex 3344 . . . . 5 𝑥 ∈ V
5 elimasng 5650 . . . . 5 ((𝐶𝑉𝑥 ∈ V) → (𝑥 ∈ ((𝐴𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵)))
64, 5mpan2 709 . . . 4 (𝐶𝑉 → (𝑥 ∈ ((𝐴𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵)))
7 elimasng 5650 . . . . . 6 ((𝐶𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
84, 7mpan2 709 . . . . 5 (𝐶𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
9 elimasng 5650 . . . . . 6 ((𝐶𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
104, 9mpan2 709 . . . . 5 (𝐶𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
118, 10anbi12d 749 . . . 4 (𝐶𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
123, 6, 113bitr4rd 301 . . 3 (𝐶𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ 𝑥 ∈ ((𝐴𝐵) “ {𝐶})))
131, 12syl5rbb 273 . 2 (𝐶𝑉 → (𝑥 ∈ ((𝐴𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))))
1413eqrdv 2759 1 (𝐶𝑉 → ((𝐴𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2140  Vcvv 3341  cin 3715  {csn 4322  cop 4328  cima 5270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280
This theorem is referenced by:  restutopopn  22264  ustuqtop2  22268
  Copyright terms: Public domain W3C validator