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 593	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
4	3	3adant3 1132	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
5		cnvimassrndm 6141	. . . . 5 ⊢ (ran 𝐺 ⊆ 𝐴 → (◡𝐺 “ 𝐴) = dom 𝐺)
6	5	3ad2ant3 1135	. . . 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 1134	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom 𝐺 = 𝐵)
9	6, 8	eqtr2d 2771	. . 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 257	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ⊆ wss 3926  ^◡ccnv 5653  dom cdm 5654  ran crn 5655   “ cima 5657   ∘ ccom 5658  Fun wfun 6524   Fn wfn 6525

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fun 6532  df-fn 6533

This theorem is referenced by:  fnfco  6742  fsplitfpar  8115  fipreima  9368  updjudhcoinlf  9944  updjudhcoinrg  9945  cshco  14853  swrdco  14854  isofn  17786  prdsinvlem  19030  prdsmgp  20109  pws1  20283  frlmbas  21713  frlmup3  21758  frlmup4  21759  evlslem1  22038  upxp  23559  uptx  23561  0vfval  30533  xppreima2  32575  psgnfzto1stlem  33057  tocycfvres1  33067  tocycfvres2  33068  cycpmfvlem  33069  cycpmfv3  33072  cycpmco2  33090  sseqfv1  34367  sseqfn  34368  sseqfv2  34372  volsupnfl  37635  ftc1anclem5  37667  ftc1anclem8  37670  choicefi  45172  fourierdlem42  46126  fcoreslem4  47043  ackvalsucsucval  48616  isofnALT  48949