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Theorem actfunsnf1o 34598
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 4171 . . . 4 (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32cbvmptv 5261 . . 3 (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
41, 3eqtri 2763 . 2 𝐹 = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5 vex 3482 . . . 4 𝑧 ∈ V
6 snex 5442 . . . 4 {⟨𝐼, 𝑘⟩} ∈ V
75, 6unex 7763 . . 3 (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
87a1i 11 . 2 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V)
9 vex 3482 . . . 4 𝑦 ∈ V
109resex 6049 . . 3 (𝑦𝐵) ∈ V
1110a1i 11 . 2 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐵) ∈ V)
12 rspe 3247 . . . . . . 7 ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
134, 7elrnmpti 5976 . . . . . . 7 (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
1412, 13sylibr 234 . . . . . 6 ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
1514adantll 714 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
16 simpr 484 . . . . . . 7 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
1716reseq1d 5999 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
18 actfunsn.1 . . . . . . . . . 10 ((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶m 𝐵))
1918sselda 3995 . . . . . . . . 9 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → 𝑧 ∈ (𝐶m 𝐵))
20 elmapfn 8904 . . . . . . . . 9 (𝑧 ∈ (𝐶m 𝐵) → 𝑧 Fn 𝐵)
2119, 20syl 17 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → 𝑧 Fn 𝐵)
22 actfunsn.3 . . . . . . . . . 10 (𝜑𝐼𝑉)
23 fnsng 6620 . . . . . . . . . 10 ((𝐼𝑉𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
2422, 23sylan 580 . . . . . . . . 9 ((𝜑𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
2524adantr 480 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
26 actfunsn.4 . . . . . . . . . . 11 (𝜑 → ¬ 𝐼𝐵)
27 disjsn 4716 . . . . . . . . . . 11 ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼𝐵)
2826, 27sylibr 234 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
2928adantr 480 . . . . . . . . 9 ((𝜑𝑘𝐶) → (𝐵 ∩ {𝐼}) = ∅)
3029adantr 480 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → (𝐵 ∩ {𝐼}) = ∅)
31 fnunres1 6681 . . . . . . . 8 ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ (𝐵 ∩ {𝐼}) = ∅) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3221, 25, 30, 31syl3anc 1370 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3332adantr 480 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3417, 33eqtr2d 2776 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 = (𝑦𝐵))
3515, 34jca 511 . . . 4 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵)))
3635anasss 466 . . 3 (((𝜑𝑘𝐶) ∧ (𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))) → (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵)))
37 simpr 484 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑧 = (𝑦𝐵))
38 simpr 484 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3938reseq1d 5999 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
4018ad3antrrr 730 . . . . . . . . . . . . 13 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐴 ⊆ (𝐶m 𝐵))
41 simplr 769 . . . . . . . . . . . . 13 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧𝐴)
4240, 41sseldd 3996 . . . . . . . . . . . 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 732 . . . . . . . . . . . 12 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐼𝑉)
45 simp-4r 784 . . . . . . . . . . . 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 732 . . . . . . . . . . 11 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝐵 ∩ {𝐼}) = ∅)
4843, 46, 47, 31syl3anc 1370 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
4948, 41eqeltrd 2839 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) ∈ 𝐴)
5039, 49eqeltrd 2839 . . . . . . . 8 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) ∈ 𝐴)
51 simpr 484 . . . . . . . . 9 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
5251, 13sylib 218 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5350, 52r19.29a 3160 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐵) ∈ 𝐴)
5453adantr 480 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑦𝐵) ∈ 𝐴)
5537, 54eqeltrd 2839 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑧𝐴)
5637uneq1d 4177 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) = ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}))
5739, 48eqtrd 2775 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = 𝑧)
5857uneq1d 4177 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5958, 38eqtr4d 2778 . . . . . . . 8 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6059, 52r19.29a 3160 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6160adantr 480 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6256, 61eqtr2d 2776 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
6355, 62jca 511 . . . 4 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
6463anasss 466 . . 3 (((𝜑𝑘𝐶) ∧ (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵))) → (𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
6536, 64impbida 801 . 2 ((𝜑𝑘𝐶) → ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) ↔ (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵))))
664, 8, 11, 65f1od 7685 1 ((𝜑𝑘𝐶) → 𝐹:𝐴1-1-onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631  cop 4637  cmpt 5231  ran crn 5690  cres 5691   Fn wfn 6558  1-1-ontowf1o 6562  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  breprexplema  34624
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