Theorem intima0 44415

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 3462	. . 3 ⊢ 𝑎 ∈ V
2	1	imaex 7920	. 2 ⊢ (𝑎 “ 𝐵) ∈ V
3	2	dfiin2 5002	1 ⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}

This proof depends on syntax axioms:   = wceq 1570  {cab 2744  ∃wrex 3092  ∩ cint 4917  ∩ ciin 4962   “ cima 5669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iin 4964  df-br 5115  df-opab 5179  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is used by:  intimass2  44422  intimasn2  44425