Theorem elmapssresd 41525

Description: A restricted mapping is a mapping. EDITORIAL: Could be used to shorten elpm2r 8834 with some reordering involving mapsspm 8865. (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 8856	. 2 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷))
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷))

Syntax hints:   → wi 4   ∈ wcel 2098   ⊆ wss 3940   ↾ cres 5668  (class class class)co 7401   ↑_m cmap 8815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8817

This theorem is referenced by:  evlselv  41614