Theorem intima0 43885

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 3444	. . 3 ⊢ 𝑎 ∈ V
2	1	imaex 7856	. 2 ⊢ (𝑎 “ 𝐵) ∈ V
3	2	dfiin2 4988	1 ⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}

Syntax hints:   = wceq 1541  {cab 2714  ∃wrex 3060  ∩ cint 4902  ∩ ciin 4947   “ cima 5627

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 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iin 4949  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  intimass2  43892  intimasn2  43895