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Theorem actfunsnf1o 32584
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 4090 . . . 4 (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32cbvmptv 5187 . . 3 (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
41, 3eqtri 2766 . 2 𝐹 = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5 vex 3436 . . . 4 𝑧 ∈ V
6 snex 5354 . . . 4 {⟨𝐼, 𝑘⟩} ∈ V
75, 6unex 7596 . . 3 (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
87a1i 11 . 2 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V)
9 vex 3436 . . . 4 𝑦 ∈ V
109resex 5939 . . 3 (𝑦𝐵) ∈ V
1110a1i 11 . 2 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐵) ∈ V)
12 rspe 3237 . . . . . . 7 ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
134, 7elrnmpti 5869 . . . . . . 7 (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
1412, 13sylibr 233 . . . . . 6 ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
1514adantll 711 . . . . 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 5890 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
18 actfunsn.1 . . . . . . . . . 10 ((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶m 𝐵))
1918sselda 3921 . . . . . . . . 9 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → 𝑧 ∈ (𝐶m 𝐵))
20 elmapfn 8653 . . . . . . . . 9 (𝑧 ∈ (𝐶m 𝐵) → 𝑧 Fn 𝐵)
2119, 20syl 17 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → 𝑧 Fn 𝐵)
22 actfunsn.3 . . . . . . . . . 10 (𝜑𝐼𝑉)
23 fnsng 6486 . . . . . . . . . 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 4647 . . . . . . . . . . 11 ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼𝐵)
2826, 27sylibr 233 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
2928adantr 481 . . . . . . . . 9 ((𝜑𝑘𝐶) → (𝐵 ∩ {𝐼}) = ∅)
3029adantr 481 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → (𝐵 ∩ {𝐼}) = ∅)
31 fnunres1 30945 . . . . . . . 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 481 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3417, 33eqtr2d 2779 . . . . 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 5890 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
4018ad3antrrr 727 . . . . . . . . . . . . 13 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐴 ⊆ (𝐶m 𝐵))
41 simplr 766 . . . . . . . . . . . . 13 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧𝐴)
4240, 41sseldd 3922 . . . . . . . . . . . 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 729 . . . . . . . . . . . 12 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐼𝑉)
45 simp-4r 781 . . . . . . . . . . . 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 729 . . . . . . . . . . 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 485 . . . . . . . . 9 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
5251, 13sylib 217 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5350, 52r19.29a 3218 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐵) ∈ 𝐴)
5453adantr 481 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑦𝐵) ∈ 𝐴)
5537, 54eqeltrd 2839 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑧𝐴)
5637uneq1d 4096 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) = ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}))
5739, 48eqtrd 2778 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = 𝑧)
5857uneq1d 4096 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5958, 38eqtr4d 2781 . . . . . . . 8 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6059, 52r19.29a 3218 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6160adantr 481 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6256, 61eqtr2d 2779 . . . . 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 798 . 2 ((𝜑𝑘𝐶) → ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) ↔ (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵))))
664, 8, 11, 65f1od 7521 1 ((𝜑𝑘𝐶) → 𝐹:𝐴1-1-onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wcel 2106  wrex 3065  Vcvv 3432  cun 3885  cin 3886  wss 3887  c0 4256  {csn 4561  cop 4567  cmpt 5157  ran crn 5590  cres 5591   Fn wfn 6428  1-1-ontowf1o 6432  (class class class)co 7275  m cmap 8615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617
This theorem is referenced by:  breprexplema  32610
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