Theorem intima0 44402

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 3458	. . 3 ⊢ 𝑎 ∈ V
2	1	imaex 7909	. 2 ⊢ (𝑎 “ 𝐵) ∈ V
3	2	dfiin2 4996	1 ⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}

This proof depends on syntax axioms:   = wceq 1569  {cab 2740  ∃wrex 3088  ∩ cint 4911  ∩ ciin 4956   “ cima 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5256  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iin 4958  df-br 5109  df-opab 5173  df-xp 5666  df-cnv 5668  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673

This theorem is used by:  intimass2  44409  intimasn2  44412