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Theorem mapco2 42703
Description: Post-composition (renaming indices) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
Hypothesis
Ref Expression
mapco2.3 𝐸 ∈ V
Assertion
Ref Expression
mapco2 ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷:𝐸𝐶) → (𝐴𝐷) ∈ (𝐵m 𝐸))

Proof of Theorem mapco2
StepHypRef Expression
1 mapco2.3 . 2 𝐸 ∈ V
2 mapco2g 42702 . 2 ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷:𝐸𝐶) → (𝐴𝐷) ∈ (𝐵m 𝐸))
31, 2mp3an1 1447 1 ((𝐴 ∈ (𝐵m 𝐶) ∧ 𝐷:𝐸𝐶) → (𝐴𝐷) ∈ (𝐵m 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Vcvv 3478  ccom 5693  wf 6559  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  diophren  42801  rabrenfdioph  42802
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