Theorem funcoressn 47043

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 5994	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ dom 𝐹 → dom (𝐹 ↾ {(𝐺‘𝑋)}) = {(𝐺‘𝑋)})
2		df-fn 6514	. . . . . . . . 9 ⊢ ((𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)} ↔ (Fun (𝐹 ↾ {(𝐺‘𝑋)}) ∧ dom (𝐹 ↾ {(𝐺‘𝑋)}) = {(𝐺‘𝑋)}))
3	2	simplbi2com 502	. . . . . . . 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 406	. . . . . 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 6940	. . . . . . . . 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 5651	. . . . . . . 8 ⊢ (𝐺 “ {𝑋}) = ran (𝐺 ↾ {𝑋})
10	8, 9	eqtrdi 2780	. . . . . . 7 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → {(𝐺‘𝑋)} = ran (𝐺 ↾ {𝑋}))
11	10	reseq2d 5950	. . . . . 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 6613	. . . . 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 232	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}))
14		fnfun 6618	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
15		funres 6558	. . . . . . . 8 ⊢ (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
16	15	funfnd 6547	. . . . . . 7 ⊢ (Fun 𝐺 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
17	14, 16	syl 17	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
18	17	adantr 480	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
19	18	adantl 481	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
20		fnresfnco 47042	. . . 4 ⊢ (((𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}) ∧ (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋})) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
21	13, 19, 20	syl2anc 584	. . 3 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
22		fnfun 6618	. . 3 ⊢ ((𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}) → Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
23	21, 22	syl 17	. 2 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
24		resco 6223	. . 3 ⊢ ((𝐹 ∘ 𝐺) ↾ {𝑋}) = (𝐹 ∘ (𝐺 ↾ {𝑋}))
25	24	funeqi 6537	. 2 ⊢ (Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}) ↔ Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
26	23, 25	sylibr 234	1 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4589  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641   ∘ ccom 5642  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511

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 5251  ax-nul 5261  ax-pr 5387

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-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by:  afvco2  47177