Theorem elmapssresd 41787

Description: A restricted mapping is a mapping. EDITORIAL: Could be used to shorten elpm2r 8860 with some reordering involving mapsspm 8891. (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 8882	. 2 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷))
4	1, 2, 3	syl2anc 582	1 ⊢ (𝜑 → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷))

Syntax hints:   → wi 4   ∈ wcel 2098   ⊆ wss 3940   ↾ cres 5674  (class class class)co 7415   ↑_m cmap 8841

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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-map 8843

This theorem is referenced by:  evlselv  41884