Theorem funcoressn 46203

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 6013	. . . . . . . 8 ⊢ ((𝐺‘𝑋) ∈ dom 𝐹 → dom (𝐹 ↾ {(𝐺‘𝑋)}) = {(𝐺‘𝑋)})
2		df-fn 6536	. . . . . . . . 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 6960	. . . . . . . . 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 5679	. . . . . . . 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 5971	. . . . . 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 6634	. . . . 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 231	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}))
14		fnfun 6639	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
15		funres 6580	. . . . . . . 8 ⊢ (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
16	15	funfnd 6569	. . . . . . 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 46202	. . . 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 583	. . 3 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
22		fnfun 6639	. . 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 6239	. . 3 ⊢ ((𝐹 ∘ 𝐺) ↾ {𝑋}) = (𝐹 ∘ (𝐺 ↾ {𝑋}))
25	24	funeqi 6559	. 2 ⊢ (Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}) ↔ Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
26	23, 25	sylibr 233	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 1533   ∈ wcel 2098  {csn 4620  dom cdm 5666  ran crn 5667   ↾ cres 5668   “ cima 5669   ∘ ccom 5670  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is referenced by:  afvco2  46335