Theorem funcoressn 41207

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 5438	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ dom 𝐹 → dom (𝐹 ↾ {(𝐺‘𝑋)}) = {(𝐺‘𝑋)})
2		df-fn 5891	. . . . . . . . 9 ⊢ ((𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)} ↔ (Fun (𝐹 ↾ {(𝐺‘𝑋)}) ∧ dom (𝐹 ↾ {(𝐺‘𝑋)}) = {(𝐺‘𝑋)}))
3	2	simplbi2com 657	. . . . . . . 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 445	. . . . . 6 ⊢ (((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → (𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)})
6	5	adantr 481	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ↾ {(𝐺‘𝑋)}) Fn {(𝐺‘𝑋)})
7		fnsnfv 6258	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋}))
8	7	adantl 482	. . . . . . . 8 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋}))
9		df-ima 5127	. . . . . . . 8 ⊢ (𝐺 “ {𝑋}) = ran (𝐺 ↾ {𝑋})
10	8, 9	syl6eq 2672	. . . . . . 7 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → {(𝐺‘𝑋)} = ran (𝐺 ↾ {𝑋}))
11	10	reseq2d 5396	. . . . . 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 5983	. . . . 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 222	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}))
14		fnfun 5988	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
15		funres 5929	. . . . . . . 8 ⊢ (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
16		funfn 5918	. . . . . . . 8 ⊢ (Fun (𝐺 ↾ {𝑋}) ↔ (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
17	15, 16	sylib 208	. . . . . . 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 481	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
20	19	adantl 482	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
21		fnresfnco 41206	. . . 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 693	. . 3 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
23		fnfun 5988	. . 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 5639	. . 3 ⊢ ((𝐹 ∘ 𝐺) ↾ {𝑋}) = (𝐹 ∘ (𝐺 ↾ {𝑋}))
26	25	funeqi 5909	. 2 ⊢ (Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}) ↔ Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
27	24, 26	sylibr 224	1 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {csn 4177  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117   ∘ ccom 5118  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  afvco2  41256