Theorem elmapssresd 41632

Description: A restricted mapping is a mapping. EDITORIAL: Could be used to shorten elpm2r 8841 with some reordering involving mapsspm 8872. (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 8863	. 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 3943   ↾ cres 5671  (class class class)co 7405   ↑_m cmap 8822

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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824

This theorem is referenced by:  evlselv  41721