Theorem intimasn2 37950

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 37949	. 2 ⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ {𝐵})})
2		intima0 37939	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝑥 “ {𝐵}) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ {𝐵})}
3	1, 2	syl6eqr 2674	1 ⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ 𝑥 ∈ 𝐴 (𝑥 “ {𝐵}))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  {csn 4177  ∩ cint 4475  ∩ ciin 4521   “ cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iin 4523  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by: (None)