Theorem fnco 6604

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 6586	. . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
2		fncofn 6603	. . . 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 6105	. . . . 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 6589	. . . . 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 2765	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → 𝐵 = (◡𝐺 “ 𝐴))
10	9	fneq2d 6580	. 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 3905  ^◡ccnv 5622  dom cdm 5623  ran crn 5624   “ cima 5626   ∘ ccom 5627  Fun wfun 6480   Fn wfn 6481

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-fun 6488  df-fn 6489

This theorem is referenced by:  fnfco  6693  fsplitfpar  8058  fipreima  9267  updjudhcoinlf  9847  updjudhcoinrg  9848  cshco  14761  swrdco  14762  isofn  17700  prdsinvlem  18946  prdsmgp  20054  pws1  20228  frlmbas  21680  frlmup3  21725  frlmup4  21726  evlslem1  22005  upxp  23526  uptx  23528  0vfval  30568  xppreima2  32608  psgnfzto1stlem  33055  tocycfvres1  33065  tocycfvres2  33066  cycpmfvlem  33067  cycpmfv3  33070  cycpmco2  33088  sseqfv1  34356  sseqfn  34357  sseqfv2  34361  volsupnfl  37644  ftc1anclem5  37676  ftc1anclem8  37679  choicefi  45178  fourierdlem42  46131  fcoreslem4  47051  ackvalsucsucval  48674  isofnALT  49017