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Theorem actfunsnrndisj 31781
 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 485 . . . . . . 7 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
21fveq1d 6671 . . . . . 6 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓𝐼) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼))
3 actfunsn.1 . . . . . . . . . . . 12 ((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶m 𝐵))
43ad2antrr 722 . . . . . . . . . . 11 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐴 ⊆ (𝐶m 𝐵))
5 simpr 485 . . . . . . . . . . 11 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧𝐴)
64, 5sseldd 3972 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧 ∈ (𝐶m 𝐵))
7 elmapfn 8424 . . . . . . . . . 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 726 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐼𝑉)
11 simpllr 772 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑘𝐶)
12 fnsng 6405 . . . . . . . . . 10 ((𝐼𝑉𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
1310, 11, 12syl2anc 584 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
14 actfunsn.4 . . . . . . . . . . 11 (𝜑 → ¬ 𝐼𝐵)
15 disjsn 4646 . . . . . . . . . . 11 ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼𝐵)
1614, 15sylibr 235 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
1716ad3antrrr 726 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → (𝐵 ∩ {𝐼}) = ∅)
18 snidg 4596 . . . . . . . . . 10 (𝐼𝑉𝐼 ∈ {𝐼})
1910, 18syl 17 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐼 ∈ {𝐼})
20 fvun2 6754 . . . . . . . . 9 ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ ((𝐵 ∩ {𝐼}) = ∅ ∧ 𝐼 ∈ {𝐼})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
218, 13, 17, 19, 20syl112anc 1368 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
22 fvsng 6940 . . . . . . . . 9 ((𝐼𝑉𝑘𝐶) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2310, 11, 22syl2anc 584 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2421, 23eqtrd 2861 . . . . . . 7 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
2524adantr 481 . . . . . 6 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = 𝑘)
262, 25eqtrd 2861 . . . . 5 (((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑓𝐼) = 𝑘)
27 simpr 485 . . . . . 6 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → 𝑓 ∈ ran 𝐹)
28 actfunsn.5 . . . . . . . 8 𝐹 = (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
29 uneq1 4136 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3029cbvmptv 5166 . . . . . . . 8 (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3128, 30eqtri 2849 . . . . . . 7 𝐹 = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32 vex 3503 . . . . . . . 8 𝑧 ∈ V
33 snex 5328 . . . . . . . 8 {⟨𝐼, 𝑘⟩} ∈ V
3432, 33unex 7462 . . . . . . 7 (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
3531, 34elrnmpti 5831 . . . . . 6 (𝑓 ∈ ran 𝐹 ↔ ∃𝑧𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3627, 35sylib 219 . . . . 5 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → ∃𝑧𝐴 𝑓 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3726, 36r19.29a 3294 . . . 4 (((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) → (𝑓𝐼) = 𝑘)
3837ralrimiva 3187 . . 3 ((𝜑𝑘𝐶) → ∀𝑓 ∈ ran 𝐹(𝑓𝐼) = 𝑘)
3938ralrimiva 3187 . 2 (𝜑 → ∀𝑘𝐶𝑓 ∈ ran 𝐹(𝑓𝐼) = 𝑘)
40 invdisj 5047 . 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 396   = wceq 1530   ∈ wcel 2107  ∀wral 3143  ∃wrex 3144  Vcvv 3500   ∪ cun 3938   ∩ cin 3939   ⊆ wss 3940  ∅c0 4295  {csn 4564  ⟨cop 4570  Disj wdisj 5028   ↦ cmpt 5143  ran crn 5555   Fn wfn 6349  ‘cfv 6354  (class class class)co 7150   ↑m cmap 8401 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-disj 5029  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-map 8403 This theorem is referenced by:  breprexplema  31806
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