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Theorem actfunsnrndisj 33218
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 6844 . . . . . 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 485 . . . . . . . . . . 11 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧𝐴)
64, 5sseldd 3945 . . . . . . . . . 10 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝑧 ∈ (𝐶m 𝐵))
7 elmapfn 8803 . . . . . . . . . 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 6553 . . . . . . . . . 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 4672 . . . . . . . . . . 11 ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼𝐵)
1614, 15sylibr 233 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
1716ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → (𝐵 ∩ {𝐼}) = ∅)
18 snidg 4620 . . . . . . . . . 10 (𝐼𝑉𝐼 ∈ {𝐼})
1910, 18syl 17 . . . . . . . . 9 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → 𝐼 ∈ {𝐼})
20 fvun2 6933 . . . . . . . . 9 ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ ((𝐵 ∩ {𝐼}) = ∅ ∧ 𝐼 ∈ {𝐼})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
218, 13, 17, 19, 20syl112anc 1374 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩})‘𝐼) = ({⟨𝐼, 𝑘⟩}‘𝐼))
22 fvsng 7126 . . . . . . . . 9 ((𝐼𝑉𝑘𝐶) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2310, 11, 22syl2anc 584 . . . . . . . 8 ((((𝜑𝑘𝐶) ∧ 𝑓 ∈ ran 𝐹) ∧ 𝑧𝐴) → ({⟨𝐼, 𝑘⟩}‘𝐼) = 𝑘)
2421, 23eqtrd 2776 . . . . . . 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 2776 . . . . 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 4116 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3029cbvmptv 5218 . . . . . . . 8 (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3128, 30eqtri 2764 . . . . . . 7 𝐹 = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32 vex 3449 . . . . . . . 8 𝑧 ∈ V
33 snex 5388 . . . . . . . 8 {⟨𝐼, 𝑘⟩} ∈ V
3432, 33unex 7680 . . . . . . 7 (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
3531, 34elrnmpti 5915 . . . . . 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 5089 . 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 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  cop 4592  Disj wdisj 5070  cmpt 5188  ran crn 5634   Fn wfn 6491  cfv 6496  (class class class)co 7357  m cmap 8765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767
This theorem is referenced by:  breprexplema  33243
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