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Theorem elmapssres 8146
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 8143 . . 3 (𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴:𝐶𝐵)
2 fssres 6306 . . 3 ((𝐴:𝐶𝐵𝐷𝐶) → (𝐴𝐷):𝐷𝐵)
31, 2sylan 577 . 2 ((𝐴 ∈ (𝐵𝑚 𝐶) ∧ 𝐷𝐶) → (𝐴𝐷):𝐷𝐵)
4 elmapex 8142 . . . . 5 (𝐴 ∈ (𝐵𝑚 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
54simpld 490 . . . 4 (𝐴 ∈ (𝐵𝑚 𝐶) → 𝐵 ∈ V)
65adantr 474 . . 3 ((𝐴 ∈ (𝐵𝑚 𝐶) ∧ 𝐷𝐶) → 𝐵 ∈ V)
74simprd 491 . . . 4 (𝐴 ∈ (𝐵𝑚 𝐶) → 𝐶 ∈ V)
8 ssexg 5028 . . . . 5 ((𝐷𝐶𝐶 ∈ V) → 𝐷 ∈ V)
98ancoms 452 . . . 4 ((𝐶 ∈ V ∧ 𝐷𝐶) → 𝐷 ∈ V)
107, 9sylan 577 . . 3 ((𝐴 ∈ (𝐵𝑚 𝐶) ∧ 𝐷𝐶) → 𝐷 ∈ V)
116, 10elmapd 8135 . 2 ((𝐴 ∈ (𝐵𝑚 𝐶) ∧ 𝐷𝐶) → ((𝐴𝐷) ∈ (𝐵𝑚 𝐷) ↔ (𝐴𝐷):𝐷𝐵))
123, 11mpbird 249 1 ((𝐴 ∈ (𝐵𝑚 𝐶) ∧ 𝐷𝐶) → (𝐴𝐷) ∈ (𝐵𝑚 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2166  Vcvv 3413  wss 3797  cres 5343  wf 6118  (class class class)co 6904  𝑚 cmap 8121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-1st 7427  df-2nd 7428  df-map 8123
This theorem is referenced by:  nn0gsumfz  18732  mdetmul  20796  mapfzcons1cl  38124  mzpcompact2lem  38157  diophin  38179  eldiophss  38181  eldioph4b  38218  mccllem  40623  iccpartres  42241  lincresunit3lem2  43115
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