Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvco2 Structured version   Visualization version   GIF version

Theorem afvco2 41759
Description: Value of a function composition, analogous to fvco2 6490. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
afvco2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Proof of Theorem afvco2
StepHypRef Expression
1 fvco2 6490 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
21adantl 469 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
3 simpll 774 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐺𝑋) ∈ dom 𝐹)
4 df-fn 6100 . . . . . . . . 9 (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
5 simpll 774 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → Fun 𝐺)
6 eleq2 2874 . . . . . . . . . . . . . 14 (𝐴 = dom 𝐺 → (𝑋𝐴𝑋 ∈ dom 𝐺))
76eqcoms 2814 . . . . . . . . . . . . 13 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
87biimpd 220 . . . . . . . . . . . 12 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
98adantl 469 . . . . . . . . . . 11 ((Fun 𝐺 ∧ dom 𝐺 = 𝐴) → (𝑋𝐴𝑋 ∈ dom 𝐺))
109imp 395 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
115, 10jca 503 . . . . . . . . 9 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
124, 11sylanb 572 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
1312adantl 469 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (Fun 𝐺𝑋 ∈ dom 𝐺))
14 dmfco 6489 . . . . . . 7 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝑋 ∈ dom (𝐹𝐺) ↔ (𝐺𝑋) ∈ dom 𝐹))
1513, 14syl 17 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝑋 ∈ dom (𝐹𝐺) ↔ (𝐺𝑋) ∈ dom 𝐹))
163, 15mpbird 248 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → 𝑋 ∈ dom (𝐹𝐺))
17 funcoressn 41655 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))
18 df-dfat 41702 . . . . . 6 ((𝐹𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})))
19 afvfundmfveq 41721 . . . . . 6 ((𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
2018, 19sylbir 226 . . . . 5 ((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
2116, 17, 20syl2anc 575 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
22 df-dfat 41702 . . . . . 6 (𝐹 defAt (𝐺𝑋) ↔ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})))
23 afvfundmfveq 41721 . . . . . 6 (𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2422, 23sylbir 226 . . . . 5 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2524adantr 468 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
262, 21, 253eqtr4d 2850 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
27 ianor 995 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ↔ (¬ (𝐺𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺𝑋)})))
2814funfni 6198 . . . . . . . . . . 11 ((𝐺 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹𝐺) ↔ (𝐺𝑋) ∈ dom 𝐹))
2928bicomd 214 . . . . . . . . . 10 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐺𝑋) ∈ dom 𝐹𝑋 ∈ dom (𝐹𝐺)))
3029notbid 309 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 ↔ ¬ 𝑋 ∈ dom (𝐹𝐺)))
3130biimpd 220 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 → ¬ 𝑋 ∈ dom (𝐹𝐺)))
32 ndmafv 41723 . . . . . . . 8 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)'''𝑋) = V)
3331, 32syl6com 37 . . . . . . 7 (¬ (𝐺𝑋) ∈ dom 𝐹 → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
34 funressnfv 41656 . . . . . . . . . . . 12 (((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ↾ {(𝐺𝑋)}))
3534ex 399 . . . . . . . . . . 11 ((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐹 ↾ {(𝐺𝑋)})))
36 afvnfundmuv 41722 . . . . . . . . . . . 12 (¬ (𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)'''𝑋) = V)
3718, 36sylnbir 322 . . . . . . . . . . 11 (¬ (𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐹𝐺)'''𝑋) = V)
3835, 37nsyl4 157 . . . . . . . . . 10 (¬ ((𝐹𝐺)'''𝑋) = V → ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐹 ↾ {(𝐺𝑋)})))
3938com12 32 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ ((𝐹𝐺)'''𝑋) = V → Fun (𝐹 ↾ {(𝐺𝑋)})))
4039con1d 141 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ Fun (𝐹 ↾ {(𝐺𝑋)}) → ((𝐹𝐺)'''𝑋) = V))
4140com12 32 . . . . . . 7 (¬ Fun (𝐹 ↾ {(𝐺𝑋)}) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4233, 41jaoi 875 . . . . . 6 ((¬ (𝐺𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺𝑋)})) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4327, 42sylbi 208 . . . . 5 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4443imp 395 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = V)
45 afvnfundmuv 41722 . . . . . . 7 𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = V)
4622, 45sylnbir 322 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = V)
4746eqcomd 2812 . . . . 5 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → V = (𝐹'''(𝐺𝑋)))
4847adantr 468 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → V = (𝐹'''(𝐺𝑋)))
4944, 48eqtrd 2840 . . 3 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
5026, 49pm2.61ian 837 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
51 eqidd 2807 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → 𝐹 = 𝐹)
524, 9sylbi 208 . . . . . 6 (𝐺 Fn 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
5352imp 395 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → 𝑋 ∈ dom 𝐺)
54 fnfun 6195 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
55 funres 6139 . . . . . . 7 (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
5654, 55syl 17 . . . . . 6 (𝐺 Fn 𝐴 → Fun (𝐺 ↾ {𝑋}))
5756adantr 468 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐺 ↾ {𝑋}))
58 df-dfat 41702 . . . . . 6 (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
59 afvfundmfveq 41721 . . . . . 6 (𝐺 defAt 𝑋 → (𝐺'''𝑋) = (𝐺𝑋))
6058, 59sylbir 226 . . . . 5 ((𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})) → (𝐺'''𝑋) = (𝐺𝑋))
6153, 57, 60syl2anc 575 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺'''𝑋) = (𝐺𝑋))
6261eqcomd 2812 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) = (𝐺'''𝑋))
6351, 62afveq12d 41716 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹'''(𝐺𝑋)) = (𝐹'''(𝐺'''𝑋)))
6450, 63eqtrd 2840 1 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2156  Vcvv 3391  {csn 4370  dom cdm 5311  cres 5313  ccom 5315  Fun wfun 6091   Fn wfn 6092  cfv 6097   defAt wdfat 41699  '''cafv 41700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-int 4670  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-fv 6105  df-aiota 41663  df-dfat 41702  df-afv 41703
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator