Theorem intimasn 43935

Description: Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
intimasn	⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})})

Distinct variable groups:   𝐴,𝑎   𝐵,𝑎,𝑥   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝑉(𝑥,𝑎)

Proof of Theorem intimasn

Dummy variables 𝑦 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1912	. 2 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦 𝐵 ∈ 𝑉)
2		r19.12sn 4676	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎))
3	2	biimprd 248	. . 3 ⊢ (𝐵 ∈ 𝑉 → (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
4	3	alimi 1813	. 2 ⊢ (∀𝑦 𝐵 ∈ 𝑉 → ∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
5		intimag 43934	. 2 ⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})})
6	1, 4, 5	3syl 18	1 ⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})})

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2713  ∀wral 3050  ∃wrex 3059  {csn 4579  ⟨cop 4585  ∩ cint 4901   “ cima 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-br 5098  df-opab 5160  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636

This theorem is referenced by:  intimasn2  43936  brtrclfv2  44005