Theorem funresfunco 41526

Description: Composition of two functions, generalization of funco 5966. (Contributed by Alexander van der Vekens, 25-Jul-2017.)

Assertion

Ref	Expression
funresfunco	⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Proof of Theorem funresfunco

Step	Hyp	Ref	Expression
1		funco 5966	. 2 ⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
2		ssid 3657	. . . . 5 ⊢ ran 𝐺 ⊆ ran 𝐺
3		cores 5676	. . . . 5 ⊢ (ran 𝐺 ⊆ ran 𝐺 → ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹 ∘ 𝐺))
4	2, 3	ax-mp 5	. . . 4 ⊢ ((𝐹 ↾ ran 𝐺) ∘ 𝐺) = (𝐹 ∘ 𝐺)
5	4	eqcomi 2660	. . 3 ⊢ (𝐹 ∘ 𝐺) = ((𝐹 ↾ ran 𝐺) ∘ 𝐺)
6	5	funeqi 5947	. 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ Fun ((𝐹 ↾ ran 𝐺) ∘ 𝐺))
7	1, 6	sylibr 224	1 ⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ⊆ wss 3607  ran crn 5144   ↾ cres 5145   ∘ ccom 5147  Fun wfun 5920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-fun 5928

This theorem is referenced by:  fnresfnco  41527