Theorem inimasn 5099

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 3355	. . 3 ⊢ (𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})) ↔ (𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})))
2		elin 3355	. . . . 5 ⊢ (⟨𝐶, 𝑥⟩ ∈ (𝐴 ∩ 𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
3	2	a1i 9	. . . 4 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐶, 𝑥⟩ ∈ (𝐴 ∩ 𝐵) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
4		vex 2774	. . . . 5 ⊢ 𝑥 ∈ V
5		elimasng 5049	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴 ∩ 𝐵)))
6	4, 5	mpan2 425	. . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ (𝐴 ∩ 𝐵)))
7		elimasng 5049	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
8	4, 7	mpan2 425	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐴 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐴))
9		elimasng 5049	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
10	4, 9	mpan2 425	. . . . 5 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ (𝐵 “ {𝐶}) ↔ ⟨𝐶, 𝑥⟩ ∈ 𝐵))
11	8, 10	anbi12d 473	. . . 4 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ (⟨𝐶, 𝑥⟩ ∈ 𝐴 ∧ ⟨𝐶, 𝑥⟩ ∈ 𝐵)))
12	3, 6, 11	3bitr4rd 221	. . 3 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ (𝐴 “ {𝐶}) ∧ 𝑥 ∈ (𝐵 “ {𝐶})) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶})))
13	1, 12	bitr2id 193	. 2 ⊢ (𝐶 ∈ 𝑉 → (𝑥 ∈ ((𝐴 ∩ 𝐵) “ {𝐶}) ↔ 𝑥 ∈ ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶}))))
14	13	eqrdv 2202	1 ⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∩ cin 3164  {csn 3632  ⟨cop 3635   “ cima 4677

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687

This theorem is referenced by: (None)