Theorem fnco 6655

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 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Proof of Theorem fnco

Step	Hyp	Ref	Expression
1		fnfun 6637	. . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
2		fncofn 6654	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
3	1, 2	sylan2 604	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
4	3	3adant3 1150	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
5		cnvimassrndm 6151	. . . . 5 ⊢ (ran 𝐺 ⊆ 𝐴 → (◡𝐺 “ 𝐴) = dom 𝐺)
6	5	3ad2ant3 1153	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (◡𝐺 “ 𝐴) = dom 𝐺)
7		fndm 6640	. . . . 5 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
8	7	3ad2ant2 1152	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom 𝐺 = 𝐵)
9	6, 8	eqtr2d 2799	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → 𝐵 = (◡𝐺 “ 𝐴))
10	9	fneq2d 6631	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴)))
11	4, 10	mpbird 260	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ⊆ wss 3906  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666   ∘ ccom 5667  Fun wfun 6532   Fn wfn 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6540  df-fn 6541

This theorem is referenced by:  fnfco  6745  fsplitfpar  8114  fipreima  9316  updjudhcoinlf  9919  updjudhcoinrg  9920  cshco  14875  swrdco  14876  isofn  17833  prdsinvlem  19116  prdsmgp  20228  pws1  20407  frlmbas  21886  frlmup3  21931  frlmup4  21932  evlslem1  22214  upxp  23761  uptx  23763  0vfval  30936  xppreima2  32974  psgnfzto1stlem  33398  tocycfvres1  33408  tocycfvres2  33409  cycpmfvlem  33410  cycpmfv3  33413  cycpmco2  33431  sseqfv1  34757  sseqfn  34758  sseqfv2  34762  volsupnfl  38294  ftc1anclem5  38326  ftc1anclem8  38329  choicefi  45897  fourierdlem42  46843  fcoreslem4  47780  ackvalsucsucval  49445  isofnALT  49786