Theorem funcoressn 39856

Description: A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)

Assertion

Ref	Expression
funcoressn	⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}))

Proof of Theorem funcoressn

Step	Hyp	Ref	Expression
1		dmressnsn 5358	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ dom 𝐹 → dom (𝐹 ↾ {(𝐺‘𝑋)}) = {(𝐺‘𝑋)})
2		df-fn 5807	. . . . . . . . 9 ⊢ ((𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)} ↔ (Fun (𝐹 ↾ {(𝐺‘𝑋)}) ∧ dom (𝐹 ↾ {(𝐺‘𝑋)}) = {(𝐺‘𝑋)}))
3	2	simplbi2com 655	. . . . . . . 8 ⊢ (dom (𝐹 ↾ {(𝐺‘𝑋)}) = {(𝐺‘𝑋)} → (Fun (𝐹 ↾ {(𝐺‘𝑋)}) → (𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)}))
4	1, 3	syl 17	. . . . . . 7 ⊢ ((𝐺‘𝑋) ∈ dom 𝐹 → (Fun (𝐹 ↾ {(𝐺‘𝑋)}) → (𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)}))
5	4	imp 444	. . . . . 6 ⊢ (((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → (𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)})
6	5	adantr 480	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)})
7		fnsnfv 6168	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋}))
8	7	adantl 481	. . . . . . . 8 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋}))
9		df-ima 5051	. . . . . . . 8 ⊢ (𝐺 “ {𝑋}) = ran (𝐺 ↾ {𝑋})
10	8, 9	syl6eq 2660	. . . . . . 7 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → {(𝐺‘𝑋)} = ran (𝐺 ↾ {𝑋}))
11	10	reseq2d 5317	. . . . . 6 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ↾ {(𝐺‘𝑋)}) = (𝐹 ↾ ran (𝐺 ↾ {𝑋})))
12	11, 10	fneq12d 5897	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)} ↔ (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋})))
13	6, 12	mpbid 221	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}))
14		fnfun 5902	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
15		funres 5843	. . . . . . . 8 ⊢ (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
16		funfn 5833	. . . . . . . 8 ⊢ (Fun (𝐺 ↾ {𝑋}) ↔ (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
17	15, 16	sylib 207	. . . . . . 7 ⊢ (Fun 𝐺 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
18	14, 17	syl 17	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
19	18	adantr 480	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
20	19	adantl 481	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
21		fnresfnco 39855	. . . 4 ⊢ (((𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}) ∧ (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋})) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
22	13, 20, 21	syl2anc 691	. . 3 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
23		fnfun 5902	. . 3 ⊢ ((𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}) → Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
24	22, 23	syl 17	. 2 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
25		resco 5556	. . 3 ⊢ ((𝐹 ∘ 𝐺) ↾ {𝑋}) = (𝐹 ∘ (𝐺 ↾ {𝑋}))
26	25	funeqi 5824	. 2 ⊢ (Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}) ↔ Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
27	24, 26	sylibr 223	1 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812

This theorem is referenced by:  afvco2  39905