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Theorem fnco 6686
Description: Composition of two functions with domains as a function with domain. (Contributed by NM, 22-May-2006.) (Proof shortened by AV, 20-Sep-2024.)
Assertion
Ref Expression
fnco ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)

Proof of Theorem fnco
StepHypRef Expression
1 fnfun 6668 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
2 fncofn 6685 . . . 4 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn (𝐺𝐴))
31, 2sylan2 593 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹𝐺) Fn (𝐺𝐴))
433adant3 1133 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn (𝐺𝐴))
5 cnvimassrndm 6172 . . . . 5 (ran 𝐺𝐴 → (𝐺𝐴) = dom 𝐺)
653ad2ant3 1136 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐺𝐴) = dom 𝐺)
7 fndm 6671 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
873ad2ant2 1135 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom 𝐺 = 𝐵)
96, 8eqtr2d 2778 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → 𝐵 = (𝐺𝐴))
109fneq2d 6662 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → ((𝐹𝐺) Fn 𝐵 ↔ (𝐹𝐺) Fn (𝐺𝐴)))
114, 10mpbird 257 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wss 3951  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  ccom 5689  Fun wfun 6555   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564
This theorem is referenced by:  fnfco  6773  fsplitfpar  8143  fipreima  9398  updjudhcoinlf  9972  updjudhcoinrg  9973  cshco  14875  swrdco  14876  isofn  17819  prdsinvlem  19067  prdsmgp  20148  pws1  20322  frlmbas  21775  frlmup3  21820  frlmup4  21821  evlslem1  22106  upxp  23631  uptx  23633  0vfval  30625  xppreima2  32661  psgnfzto1stlem  33120  tocycfvres1  33130  tocycfvres2  33131  cycpmfvlem  33132  cycpmfv3  33135  cycpmco2  33153  sseqfv1  34391  sseqfn  34392  sseqfv2  34396  volsupnfl  37672  ftc1anclem5  37704  ftc1anclem8  37707  choicefi  45205  fourierdlem42  46164  fcoreslem4  47078  ackvalsucsucval  48609
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