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Theorem actfunsnrndisj 34270
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 483 . . . . . . 7 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
21fveq1d 6904 . . . . . 6 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓𝐼) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼))
3 actfunsn.1 . . . . . . . . . . . 12 ((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶m 𝐵))
43ad2antrr 724 . . . . . . . . . . 11 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐴 ⊆ (𝐶m 𝐵))
5 simpr 483 . . . . . . . . . . 11 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧𝐴)
64, 5sseldd 3983 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧 ∈ (𝐶m 𝐵))
7 elmapfn 8890 . . . . . . . . . 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 728 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐼𝑉)
11 simpllr 774 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑘𝐶)
12 fnsng 6610 . . . . . . . . . 10 ((𝐼𝑉𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
1310, 11, 12syl2anc 582 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
14 actfunsn.4 . . . . . . . . . . 11 (𝜑 → ¬ 𝐼𝐵)
15 disjsn 4720 . . . . . . . . . . 11 ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼𝐵)
1614, 15sylibr 233 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
1716ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → (𝐵 ∩ {𝐼}) = ∅)
18 snidg 4667 . . . . . . . . . 10 (𝐼𝑉𝐼 ∈ {𝐼})
1910, 18syl 17 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐼 ∈ {𝐼})
20 fvun2 6995 . . . . . . . . 9 ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ ((𝐵 ∩ {𝐼}) = ∅ ∧ 𝐼 ∈ {𝐼})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
218, 13, 17, 19, 20syl112anc 1371 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
22 fvsng 7195 . . . . . . . . 9 ((𝐼𝑉𝑘𝐶) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2310, 11, 22syl2anc 582 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2421, 23eqtrd 2768 . . . . . . 7 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
2524adantr 479 . . . . . 6 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
262, 25eqtrd 2768 . . . . 5 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓𝐼) = 𝑘)
27 simpr 483 . . . . . 6 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → 𝑓 ∈ ran 𝐹)
28 actfunsn.5 . . . . . . . 8 𝐹 = (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
29 uneq1 4157 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3029cbvmptv 5265 . . . . . . . 8 (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3128, 30eqtri 2756 . . . . . . 7 𝐹 = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32 vex 3477 . . . . . . . 8 𝑧 ∈ V
33 snex 5437 . . . . . . . 8 {⟨𝐼, 𝑘⟩} ∈ V
3432, 33unex 7754 . . . . . . 7 (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
3531, 34elrnmpti 5966 . . . . . 6 (𝑓 ∈ ran 𝐹 ↔ ∃𝑧𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3627, 35sylib 217 . . . . 5 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → ∃𝑧𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3726, 36r19.29a 3159 . . . 4 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → (𝑓𝐼) = 𝑘)
3837ralrimiva 3143 . . 3 ((𝜑𝑘𝐶) → ∀𝑓 ∈ ran 𝐹(𝑓𝐼) = 𝑘)
3938ralrimiva 3143 . 2 (𝜑 → ∀𝑘𝐶𝑓 ∈ ran 𝐹(𝑓𝐼) = 𝑘)
40 invdisj 5136 . 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 394   = wceq 1533  wcel 2098  wral 3058  wrex 3067  Vcvv 3473  cun 3947  cin 3948  wss 3949  c0 4326  {csn 4632  cop 4638  Disj wdisj 5117  cmpt 5235  ran crn 5683   Fn wfn 6548  cfv 6553  (class class class)co 7426  m cmap 8851
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-disj 5118  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853
This theorem is referenced by:  breprexplema  34295
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