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Theorem elmapssres 8882
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 8864 . . 3 (𝐴 ∈ (𝐵m 𝐶) → 𝐴:𝐶𝐵)
2 fssres 6757 . . 3 ((𝐴:𝐶𝐵𝐷𝐶) → (𝐴𝐷):𝐷𝐵)
31, 2sylan 578 . 2 ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷𝐶) → (𝐴𝐷):𝐷𝐵)
4 elmapex 8863 . . . . 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 5318 . . . . 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 8855 . 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 3463  wss 3940  cres 5674  wf 6538  (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:  nn0gsumfz  19941  mdetmul  22541  elmapssresd  41787  mapfzcons1cl  42202  mzpcompact2lem  42235  diophin  42256  eldiophss  42258  eldioph4b  42295  tfsconcatrev  42841  mccllem  45047  iccpartres  46820  lincresunit3lem2  47659
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