Theorem mptima2 41524

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

Hypothesis

Ref	Expression
mptima2.1	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
mptima2	⊢ (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 mptima2

Step	Hyp	Ref	Expression
1		mptima 5943	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵)
2		mptima2.1	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
3		sseqin2 4194	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
4	2, 3	sylib 220	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = 𝐶)
5	4	mpteq1d 5157	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵))
6	5	rneqd 5810	. 2 ⊢ (𝜑 → ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))
7	1, 6	syl5eq 2870	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3937   ⊆ wss 3938   ↦ cmpt 5148  ran crn 5558   “ cima 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-mpt 5149  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570

This theorem is referenced by:  limsupresico  41988  limsupvaluz  41996  liminfresico  42059