Theorem mapco2 42409

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 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸))

Proof of Theorem mapco2

Step	Hyp	Ref	Expression
1		mapco2.3	. 2 ⊢ 𝐸 ∈ V
2		mapco2g 42408	. 2 ⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸))
3	1, 2	mp3an1 1445	1 ⊢ ((𝐴 ∈ (𝐵 ↑m 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑m 𝐸))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2099  Vcvv 3462   ∘ ccom 5678  ⟶wf 6542  (class class class)co 7416   ↑_m cmap 8847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-map 8849

This theorem is referenced by:  diophren  42507  rabrenfdioph  42508