Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  actfunsnrndisj Structured version   Visualization version   GIF version

Theorem actfunsnrndisj 30811
 Description: The action 𝐹 of extending function from 𝐵 to 𝐶 with new values at point 𝐼 yields different functions. (Contributed by Thierry Arnoux, 9-Dec-2021.)
Hypotheses
Ref Expression
actfunsn.1 ((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶𝑚 𝐵))
actfunsn.2 (𝜑𝐶 ∈ V)
actfunsn.3 (𝜑𝐼𝑉)
actfunsn.4 (𝜑 → ¬ 𝐼𝐵)
actfunsn.5 𝐹 = (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
Assertion
Ref Expression
actfunsnrndisj (𝜑Disj 𝑘𝐶 ran 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑘,𝐼,𝑥   𝜑,𝑘
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑘)   𝐵(𝑥,𝑘)   𝐶(𝑥,𝑘)   𝐹(𝑥,𝑘)   𝑉(𝑥,𝑘)

Proof of Theorem actfunsnrndisj
Dummy variables 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . . . 7 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
21fveq1d 6231 . . . . . 6 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓𝐼) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼))
3 actfunsn.1 . . . . . . . . . . . 12 ((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶𝑚 𝐵))
43ad2antrr 762 . . . . . . . . . . 11 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐴 ⊆ (𝐶𝑚 𝐵))
5 simpr 476 . . . . . . . . . . 11 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧𝐴)
64, 5sseldd 3637 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧 ∈ (𝐶𝑚 𝐵))
7 elmapfn 7922 . . . . . . . . . 10 (𝑧 ∈ (𝐶𝑚 𝐵) → 𝑧 Fn 𝐵)
86, 7syl 17 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧 Fn 𝐵)
9 actfunsn.3 . . . . . . . . . . 11 (𝜑𝐼𝑉)
109ad3antrrr 766 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐼𝑉)
11 simpllr 815 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑘𝐶)
12 fnsng 5976 . . . . . . . . . 10 ((𝐼𝑉𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
1310, 11, 12syl2anc 694 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
14 actfunsn.4 . . . . . . . . . . 11 (𝜑 → ¬ 𝐼𝐵)
15 disjsn 4278 . . . . . . . . . . 11 ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼𝐵)
1614, 15sylibr 224 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
1716ad3antrrr 766 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → (𝐵 ∩ {𝐼}) = ∅)
18 snidg 4239 . . . . . . . . . 10 (𝐼𝑉𝐼 ∈ {𝐼})
1910, 18syl 17 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐼 ∈ {𝐼})
20 fvun2 6309 . . . . . . . . 9 ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ ((𝐵 ∩ {𝐼}) = ∅ ∧ 𝐼 ∈ {𝐼})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
218, 13, 17, 19, 20syl112anc 1370 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
22 fvsng 6488 . . . . . . . . 9 ((𝐼𝑉𝑘𝐶) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2310, 11, 22syl2anc 694 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2421, 23eqtrd 2685 . . . . . . 7 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
2524adantr 480 . . . . . 6 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
262, 25eqtrd 2685 . . . . 5 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓𝐼) = 𝑘)
27 simpr 476 . . . . . 6 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → 𝑓 ∈ ran 𝐹)
28 actfunsn.5 . . . . . . . 8 𝐹 = (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
29 uneq1 3793 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3029cbvmptv 4783 . . . . . . . 8 (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3128, 30eqtri 2673 . . . . . . 7 𝐹 = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32 vex 3234 . . . . . . . 8 𝑧 ∈ V
33 snex 4938 . . . . . . . 8 {⟨𝐼, 𝑘⟩} ∈ V
3432, 33unex 6998 . . . . . . 7 (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
3531, 34elrnmpti 5408 . . . . . 6 (𝑓 ∈ ran 𝐹 ↔ ∃𝑧𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3627, 35sylib 208 . . . . 5 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → ∃𝑧𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3726, 36r19.29a 3107 . . . 4 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → (𝑓𝐼) = 𝑘)
3837ralrimiva 2995 . . 3 ((𝜑𝑘𝐶) → ∀𝑓 ∈ ran 𝐹(𝑓𝐼) = 𝑘)
3938ralrimiva 2995 . 2 (𝜑 → ∀𝑘𝐶𝑓 ∈ ran 𝐹(𝑓𝐼) = 𝑘)
40 invdisj 4670 . 2 (∀𝑘𝐶𝑓 ∈ ran 𝐹(𝑓𝐼) = 𝑘Disj 𝑘𝐶 ran 𝐹)
4139, 40syl 17 1 (𝜑Disj 𝑘𝐶 ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {csn 4210  ⟨cop 4216  Disj wdisj 4652   ↦ cmpt 4762  ran crn 5144   Fn wfn 5921  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901 This theorem is referenced by:  breprexplema  30836
 Copyright terms: Public domain W3C validator