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Theorem elmapssres 8886
Description: A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
elmapssres ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷𝐶) → (𝐴𝐷) ∈ (𝐵m 𝐷))

Proof of Theorem elmapssres
StepHypRef Expression
1 elmapi 8868 . . 3 (𝐴 ∈ (𝐵m 𝐶) → 𝐴:𝐶𝐵)
2 fssres 6763 . . 3 ((𝐴:𝐶𝐵𝐷𝐶) → (𝐴𝐷):𝐷𝐵)
31, 2sylan 578 . 2 ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷𝐶) → (𝐴𝐷):𝐷𝐵)
4 elmapex 8867 . . . . 5 (𝐴 ∈ (𝐵m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
54simpld 493 . . . 4 (𝐴 ∈ (𝐵m 𝐶) → 𝐵 ∈ V)
65adantr 479 . . 3 ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷𝐶) → 𝐵 ∈ V)
74simprd 494 . . . 4 (𝐴 ∈ (𝐵m 𝐶) → 𝐶 ∈ V)
8 ssexg 5324 . . . . 5 ((𝐷𝐶𝐶 ∈ V) → 𝐷 ∈ V)
98ancoms 457 . . . 4 ((𝐶 ∈ V ∧ 𝐷𝐶) → 𝐷 ∈ V)
107, 9sylan 578 . . 3 ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷𝐶) → 𝐷 ∈ V)
116, 10elmapd 8859 . 2 ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷𝐶) → ((𝐴𝐷) ∈ (𝐵m 𝐷) ↔ (𝐴𝐷):𝐷𝐵))
123, 11mpbird 256 1 ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷𝐶) → (𝐴𝐷) ∈ (𝐵m 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  Vcvv 3461  wss 3944  cres 5680  wf 6545  (class class class)co 7419  m cmap 8845
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 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
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 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847
This theorem is referenced by:  nn0gsumfz  19951  mdetmul  22569  elmapssresd  41865  mapfzcons1cl  42280  mzpcompact2lem  42313  diophin  42334  eldiophss  42336  eldioph4b  42373  tfsconcatrev  42919  mccllem  45123  iccpartres  46895  lincresunit3lem2  47734
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