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Theorem mapco2 37097
 Description: Post-composition (renaming indexes) 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 37096 . 2 ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵𝑚 𝐶) ∧ 𝐷:𝐸𝐶) → (𝐴𝐷) ∈ (𝐵𝑚 𝐸))
31, 2mp3an1 1409 1 ((𝐴 ∈ (𝐵𝑚 𝐶) ∧ 𝐷:𝐸𝐶) → (𝐴𝐷) ∈ (𝐵𝑚 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1988  Vcvv 3195   ∘ ccom 5108  ⟶wf 5872  (class class class)co 6635   ↑𝑚 cmap 7842 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844 This theorem is referenced by:  diophren  37196  rabrenfdioph  37197
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