Theorem elmapssresd 42260

Description: A restricted mapping is a mapping. EDITORIAL: Could be used to shorten elpm2r 8883 with some reordering involving mapsspm 8914. (Contributed by SN, 11-Mar-2025.)

Hypotheses

Ref	Expression
elmapssresd.1	⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m 𝐶))
elmapssresd.2	⊢ (𝜑 → 𝐷 ⊆ 𝐶)

Assertion

Ref	Expression
elmapssresd	⊢ (𝜑 → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷))

Proof of Theorem elmapssresd

Step	Hyp	Ref	Expression
1		elmapssresd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m 𝐶))
2		elmapssresd.2	. 2 ⊢ (𝜑 → 𝐷 ⊆ 𝐶)
3		elmapssres 8905	. 2 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷))
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷))

Syntax hints:   → wi 4   ∈ wcel 2105   ⊆ wss 3962   ↾ cres 5690  (class class class)co 7430   ↑_m cmap 8864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866

This theorem is referenced by:  evlselv  42573