Theorem inimasn 5469

Description: The intersection of the image of singleton. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
inimasn	⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))

Proof of Theorem inimasn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3758	. . 3 ⊢ (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})))
2		elin 3758	. . . . 5 ⊢ (⟨𝐶, 𝑥⟩ ∈ (𝐴 ∩ 𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
3	2	a1i 11	. . . 4 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐶, 𝑥⟩ ∈ (𝐴 ∩ 𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
4		vex 3176	. . . . 5 ⊢ 𝑥 ∈ V
5		elimasng 5410	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴 ∩ 𝐵)))
6	4, 5	mpan2 703	. . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴 ∩ 𝐵)))
7		elimasng 5410	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
8	4, 7	mpan2 703	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
9		elimasng 5410	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
10	4, 9	mpan2 703	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
11	8, 10	anbi12d 743	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
12	3, 6, 11	3bitr4rd 300	. . 3 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶})))
13	1, 12	syl5rbb 272	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))))
14	13	eqrdv 2608	1 ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  {csn 4125  ⟨cop 4131   “ cima 5041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051

This theorem is referenced by:  restutopopn  21852  ustuqtop2  21856