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Theorem inimasn 6012
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 4168 . . 3 (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})))
2 elin 4168 . . . . 5 (⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
32a1i 11 . . . 4 (𝐶𝑉 → (⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
4 elimasng 5954 . . . . 5 ((𝐶𝑉𝑥 ∈ V) → (𝑥 ∈ ((𝐴𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵)))
54elvd 3500 . . . 4 (𝐶𝑉 → (𝑥 ∈ ((𝐴𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴𝐵)))
6 elimasng 5954 . . . . . 6 ((𝐶𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
76elvd 3500 . . . . 5 (𝐶𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
8 elimasng 5954 . . . . . 6 ((𝐶𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
98elvd 3500 . . . . 5 (𝐶𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
107, 9anbi12d 632 . . . 4 (𝐶𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
113, 5, 103bitr4rd 314 . . 3 (𝐶𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ 𝑥 ∈ ((𝐴𝐵) “ {𝐶})))
121, 11syl5rbb 286 . 2 (𝐶𝑉 → (𝑥 ∈ ((𝐴𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))))
1312eqrdv 2819 1 (𝐶𝑉 → ((𝐴𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  Vcvv 3494  cin 3934  {csn 4566  cop 4572  cima 5557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567
This theorem is referenced by:  restutopopn  22846  ustuqtop2  22850
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