Theorem intima0 43654

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 3485	. . 3 ⊢ 𝑎 ∈ V
2	1	imaex 7944	. 2 ⊢ (𝑎 “ 𝐵) ∈ V
3	2	dfiin2 5042	1 ⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}

Syntax hints:   = wceq 1539  {cab 2714  ∃wrex 3070  ∩ cint 4954  ∩ ciin 5000   “ cima 5696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-int 4955  df-iin 5002  df-br 5152  df-opab 5214  df-xp 5699  df-cnv 5701  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706

This theorem is referenced by:  intimass2  43661  intimasn2  43664