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Theorem actfunsnf1o 31761
Description: The action 𝐹 of extending function from 𝐵 to 𝐶 with new values at point 𝐼 is a bijection. (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
actfunsnf1o ((𝜑𝑘𝐶) → 𝐹:𝐴1-1-onto→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 actfunsnf1o
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 actfunsn.5 . . 3 𝐹 = (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
2 uneq1 4136 . . . 4 (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32cbvmptv 5166 . . 3 (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
41, 3eqtri 2849 . 2 𝐹 = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5 vex 3503 . . . 4 𝑧 ∈ V
6 snex 5328 . . . 4 {⟨𝐼, 𝑘⟩} ∈ V
75, 6unex 7458 . . 3 (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
87a1i 11 . 2 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V)
9 vex 3503 . . . 4 𝑦 ∈ V
109resex 5898 . . 3 (𝑦𝐵) ∈ V
1110a1i 11 . 2 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐵) ∈ V)
12 rspe 3309 . . . . . . 7 ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
134, 7elrnmpti 5831 . . . . . . 7 (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
1412, 13sylibr 235 . . . . . 6 ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
1514adantll 710 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
16 simpr 485 . . . . . . 7 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
1716reseq1d 5851 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
18 actfunsn.1 . . . . . . . . . 10 ((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶m 𝐵))
1918sselda 3971 . . . . . . . . 9 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → 𝑧 ∈ (𝐶m 𝐵))
20 elmapfn 8419 . . . . . . . . 9 (𝑧 ∈ (𝐶m 𝐵) → 𝑧 Fn 𝐵)
2119, 20syl 17 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → 𝑧 Fn 𝐵)
22 actfunsn.3 . . . . . . . . . 10 (𝜑𝐼𝑉)
23 fnsng 6403 . . . . . . . . . 10 ((𝐼𝑉𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
2422, 23sylan 580 . . . . . . . . 9 ((𝜑𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
2524adantr 481 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
26 actfunsn.4 . . . . . . . . . . 11 (𝜑 → ¬ 𝐼𝐵)
27 disjsn 4646 . . . . . . . . . . 11 ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼𝐵)
2826, 27sylibr 235 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
2928adantr 481 . . . . . . . . 9 ((𝜑𝑘𝐶) → (𝐵 ∩ {𝐼}) = ∅)
3029adantr 481 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → (𝐵 ∩ {𝐼}) = ∅)
31 fnunres1 30271 . . . . . . . 8 ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ (𝐵 ∩ {𝐼}) = ∅) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3221, 25, 30, 31syl3anc 1365 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3332adantr 481 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3417, 33eqtr2d 2862 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 = (𝑦𝐵))
3515, 34jca 512 . . . 4 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵)))
3635anasss 467 . . 3 (((𝜑𝑘𝐶) ∧ (𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))) → (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵)))
37 simpr 485 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑧 = (𝑦𝐵))
38 simpr 485 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3938reseq1d 5851 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
4018ad3antrrr 726 . . . . . . . . . . . . 13 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐴 ⊆ (𝐶m 𝐵))
41 simplr 765 . . . . . . . . . . . . 13 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧𝐴)
4240, 41sseldd 3972 . . . . . . . . . . . 12 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 ∈ (𝐶m 𝐵))
4342, 20syl 17 . . . . . . . . . . 11 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 Fn 𝐵)
4422ad4antr 728 . . . . . . . . . . . 12 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐼𝑉)
45 simp-4r 780 . . . . . . . . . . . 12 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑘𝐶)
4644, 45, 23syl2anc 584 . . . . . . . . . . 11 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
4728ad4antr 728 . . . . . . . . . . 11 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝐵 ∩ {𝐼}) = ∅)
4843, 46, 47, 31syl3anc 1365 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
4948, 41eqeltrd 2918 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) ∈ 𝐴)
5039, 49eqeltrd 2918 . . . . . . . 8 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) ∈ 𝐴)
51 simpr 485 . . . . . . . . 9 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
5251, 13sylib 219 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5350, 52r19.29a 3294 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐵) ∈ 𝐴)
5453adantr 481 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑦𝐵) ∈ 𝐴)
5537, 54eqeltrd 2918 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑧𝐴)
5637uneq1d 4142 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) = ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}))
5739, 48eqtrd 2861 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = 𝑧)
5857uneq1d 4142 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5958, 38eqtr4d 2864 . . . . . . . 8 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6059, 52r19.29a 3294 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6160adantr 481 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6256, 61eqtr2d 2862 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
6355, 62jca 512 . . . 4 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
6463anasss 467 . . 3 (((𝜑𝑘𝐶) ∧ (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵))) → (𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
6536, 64impbida 797 . 2 ((𝜑𝑘𝐶) → ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) ↔ (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵))))
664, 8, 11, 65f1od 7387 1 ((𝜑𝑘𝐶) → 𝐹:𝐴1-1-onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107  wrex 3144  Vcvv 3500  cun 3938  cin 3939  wss 3940  c0 4295  {csn 4564  cop 4570  cmpt 5143  ran crn 5555  cres 5556   Fn wfn 6347  1-1-ontowf1o 6351  (class class class)co 7148  m cmap 8396
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 7451
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-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-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 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-map 8398
This theorem is referenced by:  breprexplema  31787
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