Theorem mptima 6071

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 5689	. 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 5936	. 2 ⊢ ran ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
4	1, 3	eqtri 2760	1 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)

Syntax hints:   = wceq 1541   ∩ cin 3947   ↦ cmpt 5231  ran crn 5677   ↾ cres 5678   “ cima 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689

This theorem is referenced by:  mptimass  6072  fsplitfpar  8103  elmptima  43952