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Theorem fnco 6667
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 6649 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
2 fncofn 6666 . . . 4 ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹𝐺) Fn (𝐺𝐴))
31, 2sylan2 592 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹𝐺) Fn (𝐺𝐴))
433adant3 1131 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn (𝐺𝐴))
5 cnvimassrndm 6151 . . . . 5 (ran 𝐺𝐴 → (𝐺𝐴) = dom 𝐺)
653ad2ant3 1134 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐺𝐴) = dom 𝐺)
7 fndm 6652 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
873ad2ant2 1133 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom 𝐺 = 𝐵)
96, 8eqtr2d 2772 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → 𝐵 = (𝐺𝐴))
109fneq2d 6643 . 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 1086   = wceq 1540  wss 3948  ccnv 5675  dom cdm 5676  ran crn 5677  cima 5679  ccom 5680  Fun wfun 6537   Fn wfn 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546
This theorem is referenced by:  fcoOLD  6742  fnfco  6756  fsplitfpar  8109  fipreima  9364  updjudhcoinlf  9933  updjudhcoinrg  9934  cshco  14794  swrdco  14795  isofn  17729  prdsinvlem  18975  prdsmgp  20052  pws1  20220  frlmbas  21620  frlmup3  21665  frlmup4  21666  evlslem1  21956  upxp  23447  uptx  23449  0vfval  30292  xppreima2  32309  psgnfzto1stlem  32695  tocycfvres1  32705  tocycfvres2  32706  cycpmfvlem  32707  cycpmfv3  32710  cycpmco2  32728  sseqfv1  33852  sseqfn  33853  sseqfv2  33857  volsupnfl  36997  ftc1anclem5  37029  ftc1anclem8  37032  choicefi  44358  fourierdlem42  45324  fcoreslem4  46235  ackvalsucsucval  47536
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