Theorem mptima 6072

Description: Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

Ref	Expression
mptima	⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem mptima

Step	Hyp	Ref	Expression
1		df-ima 5680	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶)
2		resmpt3 6038	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
3	2	rneqi 5930	. 2 ⊢ ran ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
4	1, 3	eqtri 2757	1 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)

Syntax hints:   = wceq 1539   ∩ cin 3932   ↦ cmpt 5207  ran crn 5668   ↾ cres 5669   “ cima 5670

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 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-br 5126  df-opab 5188  df-mpt 5208  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680

This theorem is referenced by:  mptimass  6073  fsplitfpar  8126  elmptima  45210