MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fncofn Structured version   Visualization version   GIF version

Theorem fncofn 6659
Description: Composition of a function with domain and a function as a function with domain. Generalization of fnco 6660. (Contributed by AV, 17-Sep-2024.)
Assertion
Ref Expression
fncofn ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn (𝐺𝐴))

Proof of Theorem fncofn
StepHypRef Expression
1 fnfun 6642 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 funco 6581 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
31, 2sylan 579 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → Fun (𝐹𝐺))
43funfnd 6572 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn dom (𝐹𝐺))
5 fndm 6645 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 480 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → dom 𝐹 = 𝐴)
76eqcomd 2732 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → 𝐴 = dom 𝐹)
87imaeq2d 6052 . . . 4 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = (𝐺 “ dom 𝐹))
9 dmco 6246 . . . 4 dom (𝐹𝐺) = (𝐺 “ dom 𝐹)
108, 9eqtr4di 2784 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐺𝐴) = dom (𝐹𝐺))
1110fneq2d 6636 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → ((𝐹𝐺) Fn (𝐺𝐴) ↔ (𝐹𝐺) Fn dom (𝐹𝐺)))
124, 11mpbird 257 1 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  ccnv 5668  dom cdm 5669  cima 5672  ccom 5673  Fun wfun 6530   Fn wfn 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fun 6538  df-fn 6539
This theorem is referenced by:  fnco  6660  fcof  6733
  Copyright terms: Public domain W3C validator