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Theorem actfunsnrndisj 34173
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 484 . . . . . . 7 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
21fveq1d 6893 . . . . . 6 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓𝐼) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼))
3 actfunsn.1 . . . . . . . . . . . 12 ((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶m 𝐵))
43ad2antrr 725 . . . . . . . . . . 11 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐴 ⊆ (𝐶m 𝐵))
5 simpr 484 . . . . . . . . . . 11 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧𝐴)
64, 5sseldd 3979 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧 ∈ (𝐶m 𝐵))
7 elmapfn 8875 . . . . . . . . . 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 729 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐼𝑉)
11 simpllr 775 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑘𝐶)
12 fnsng 6599 . . . . . . . . . 10 ((𝐼𝑉𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
1310, 11, 12syl2anc 583 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
14 actfunsn.4 . . . . . . . . . . 11 (𝜑 → ¬ 𝐼𝐵)
15 disjsn 4711 . . . . . . . . . . 11 ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼𝐵)
1614, 15sylibr 233 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
1716ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → (𝐵 ∩ {𝐼}) = ∅)
18 snidg 4658 . . . . . . . . . 10 (𝐼𝑉𝐼 ∈ {𝐼})
1910, 18syl 17 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐼 ∈ {𝐼})
20 fvun2 6984 . . . . . . . . 9 ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ ((𝐵 ∩ {𝐼}) = ∅ ∧ 𝐼 ∈ {𝐼})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
218, 13, 17, 19, 20syl112anc 1372 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
22 fvsng 7183 . . . . . . . . 9 ((𝐼𝑉𝑘𝐶) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2310, 11, 22syl2anc 583 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2421, 23eqtrd 2767 . . . . . . 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 2767 . . . . 5 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓𝐼) = 𝑘)
27 simpr 484 . . . . . 6 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → 𝑓 ∈ ran 𝐹)
28 actfunsn.5 . . . . . . . 8 𝐹 = (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
29 uneq1 4152 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3029cbvmptv 5255 . . . . . . . 8 (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3128, 30eqtri 2755 . . . . . . 7 𝐹 = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32 vex 3473 . . . . . . . 8 𝑧 ∈ V
33 snex 5427 . . . . . . . 8 {⟨𝐼, 𝑘⟩} ∈ V
3432, 33unex 7742 . . . . . . 7 (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
3531, 34elrnmpti 5956 . . . . . 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 3157 . . . 4 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → (𝑓𝐼) = 𝑘)
3837ralrimiva 3141 . . 3 ((𝜑𝑘𝐶) → ∀𝑓 ∈ ran 𝐹(𝑓𝐼) = 𝑘)
3938ralrimiva 3141 . 2 (𝜑 → ∀𝑘𝐶𝑓 ∈ ran 𝐹(𝑓𝐼) = 𝑘)
40 invdisj 5126 . 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 395   = wceq 1534  wcel 2099  wral 3056  wrex 3065  Vcvv 3469  cun 3942  cin 3943  wss 3944  c0 4318  {csn 4624  cop 4630  Disj wdisj 5107  cmpt 5225  ran crn 5673   Fn wfn 6537  cfv 6542  (class class class)co 7414  m cmap 8836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8838
This theorem is referenced by:  breprexplema  34198
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