Theorem fnco 6649

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 6631	. . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
2		fncofn 6648	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
3	1, 2	sylan2 605	. . 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 6141	. . . . 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 6634	. . . . 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 2797	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → 𝐵 = (◡𝐺 “ 𝐴))
10	9	fneq2d 6625	. 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)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ⊆ wss 3899  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654   ∘ ccom 5655  Fun wfun 6525   Fn wfn 6526

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6533  df-fn 6534

This theorem is used by:  fnfco  6739  fsplitfpar  8118  fipreima  9331  updjudhcoinlf  9994  updjudhcoinrg  9995  cshco  14967  swrdco  14968  isofn  17930  prdsinvlem  19239  prdsmgp  20351  pws1  20534  frlmbas  22041  frlmup3  22086  frlmup4  22087  evlslem1  22371  upxp  23922  uptx  23924  0vfval  31190  xppreima2  33227  psgnfzto1stlem  33643  tocycfvres1  33653  tocycfvres2  33654  cycpmfvlem  33655  cycpmfv3  33658  cycpmco2  33676  sseqfv1  35004  sseqfn  35005  sseqfv2  35009  volsupnfl  38551  ftc1anclem5  38583  ftc1anclem8  38586  choicefi  46157  fourierdlem42  47103  fcoreslem4  48080  ackvalsucsucval  49744  isofnALT  50083