Theorem intimasn2 41155

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
intimasn2	⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ 𝑥 ∈ 𝐴 (𝑥 “ {𝐵}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem intimasn2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intimasn 41154	. 2 ⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ {𝐵})})
2		intima0 41145	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝑥 “ {𝐵}) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ {𝐵})}
3	1, 2	eqtr4di 2797	1 ⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ 𝑥 ∈ 𝐴 (𝑥 “ {𝐵}))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  {csn 4558  ∩ cint 4876  ∩ ciin 4922   “ cima 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iin 4924  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593

This theorem is referenced by: (None)