Theorem intima0 43687

Description: Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
intima0	⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}

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

Allowed substitution hints:   𝐴(𝑎)   𝐵(𝑎)

Proof of Theorem intima0

Step	Hyp	Ref	Expression
1		vex 3440	. . 3 ⊢ 𝑎 ∈ V
2	1	imaex 7844	. 2 ⊢ (𝑎 “ 𝐵) ∈ V
3	2	dfiin2 4983	1 ⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}

Syntax hints:   = wceq 1541  {cab 2709  ∃wrex 3056  ∩ cint 4897  ∩ ciin 4942   “ cima 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iin 4944  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629

This theorem is referenced by:  intimass2  43694  intimasn2  43697