Theorem fnco 6465

Description: Composition of two functions. (Contributed by NM, 22-May-2006.)

Assertion

Ref	Expression
fnco	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Proof of Theorem fnco

Step	Hyp	Ref	Expression
1		fnfun 6453	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fnfun 6453	. . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
3		funco 6395	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
4	1, 2, 3	syl2an 597	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → Fun (𝐹 ∘ 𝐺))
5	4	3adant3 1128	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → Fun (𝐹 ∘ 𝐺))
6		fndm 6455	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	sseq2d 3999	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺 ⊆ 𝐴))
8	7	biimpar 480	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → ran 𝐺 ⊆ dom 𝐹)
9		dmcosseq 5844	. . . . 5 ⊢ (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹 ∘ 𝐺) = dom 𝐺)
10	8, 9	syl 17	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺)
11	10	3adant2 1127	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺)
12		fndm 6455	. . . 4 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
13	12	3ad2ant2 1130	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom 𝐺 = 𝐵)
14	11, 13	eqtrd 2856	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = 𝐵)
15		df-fn 6358	. 2 ⊢ ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = 𝐵))
16	5, 14, 15	sylanbrc 585	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ⊆ wss 3936  dom cdm 5555  ran crn 5556   ∘ ccom 5559  Fun wfun 6349   Fn wfn 6350

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-fun 6357  df-fn 6358

This theorem is referenced by:  fco  6531  fnfco  6543  fsplitfpar  7814  fipreima  8830  updjudhcoinlf  9361  updjudhcoinrg  9362  cshco  14198  swrdco  14199  isofn  17045  prdsinvlem  18208  prdsmgp  19360  pws1  19366  evlslem1  20295  frlmbas  20899  frlmup3  20944  frlmup4  20945  upxp  22231  uptx  22233  0vfval  28383  xppreima2  30395  psgnfzto1stlem  30742  tocycfvres1  30752  tocycfvres2  30753  cycpmfvlem  30754  cycpmfv3  30757  cycpmco2  30775  sseqfv1  31647  sseqfn  31648  sseqfv2  31652  volsupnfl  34952  ftc1anclem5  34986  ftc1anclem8  34989  choicefi  41483  fourierdlem42  42454