Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mapcod Structured version   Visualization version   GIF version

Theorem mapcod 42263
Description: Compose two mappings. (Contributed by SN, 11-Mar-2025.)
Hypotheses
Ref Expression
mapcod.1 (𝜑𝐹 ∈ (𝐴m 𝐵))
mapcod.2 (𝜑𝐺 ∈ (𝐵m 𝐶))
Assertion
Ref Expression
mapcod (𝜑 → (𝐹𝐺) ∈ (𝐴m 𝐶))

Proof of Theorem mapcod
StepHypRef Expression
1 mapcod.1 . . . 4 (𝜑𝐹 ∈ (𝐴m 𝐵))
2 elmapex 8887 . . . 4 (𝐹 ∈ (𝐴m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2syl 17 . . 3 (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
43simpld 494 . 2 (𝜑𝐴 ∈ V)
5 mapcod.2 . . . 4 (𝜑𝐺 ∈ (𝐵m 𝐶))
6 elmapex 8887 . . . 4 (𝐺 ∈ (𝐵m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
75, 6syl 17 . . 3 (𝜑 → (𝐵 ∈ V ∧ 𝐶 ∈ V))
87simprd 495 . 2 (𝜑𝐶 ∈ V)
9 elmapi 8888 . . . 4 (𝐹 ∈ (𝐴m 𝐵) → 𝐹:𝐵𝐴)
101, 9syl 17 . . 3 (𝜑𝐹:𝐵𝐴)
11 elmapi 8888 . . . 4 (𝐺 ∈ (𝐵m 𝐶) → 𝐺:𝐶𝐵)
125, 11syl 17 . . 3 (𝜑𝐺:𝐶𝐵)
1310, 12fcod 6762 . 2 (𝜑 → (𝐹𝐺):𝐶𝐴)
144, 8, 13elmapdd 8880 1 (𝜑 → (𝐹𝐺) ∈ (𝐴m 𝐶))
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:  evlselv  42574
  Copyright terms: Public domain W3C validator