Theorem mptimass 6044

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

Hypothesis

Ref	Expression
mptimass.1	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
mptimass	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem mptimass

Step	Hyp	Ref	Expression
1		mptima 6043	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
2		mptimass.1	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
3		sseqin2 4186	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
4	2, 3	sylib 218	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = 𝐶)
5	4	mpteq1d 5197	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵))
6	5	rneqd 5902	. 2 ⊢ (𝜑 → ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))
7	1, 6	eqtrid 2776	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3913   ⊆ wss 3914   ↦ cmpt 5188  ran crn 5639   “ cima 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-mpt 5189  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651

This theorem is referenced by:  pzriprnglem10  21400  limsupresico  45698  limsupvaluz  45706  liminfresico  45769