Theorem mptimass 6032

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 6031	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
2		mptimass.1	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
3		sseqin2 4159	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
4	2, 3	sylib 219	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = 𝐶)
5	4	mpteq1d 5169	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵))
6	5	rneqd 5887	. 2 ⊢ (𝜑 → ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))
7	1, 6	eqtrid 2787	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))

Syntax hints:   → wi 4   = wceq 1547   ∩ cin 3889   ⊆ wss 3890   ↦ cmpt 5160  ran crn 5626   “ cima 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-mpt 5161  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638

This theorem is referenced by:  pzriprnglem10  21472  limsupresico  46150  limsupvaluz  46158  liminfresico  46221