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Theorem actfunsnf1o 31133
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 3922 . . . 4 (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
32cbvmptv 4909 . . 3 (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
41, 3eqtri 2787 . 2 𝐹 = (𝑧𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5 vex 3353 . . . 4 𝑧 ∈ V
6 snex 5064 . . . 4 {⟨𝐼, 𝑘⟩} ∈ V
75, 6unex 7154 . . 3 (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
87a1i 11 . 2 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V)
9 vex 3353 . . . 4 𝑦 ∈ V
109resex 5620 . . 3 (𝑦𝐵) ∈ V
1110a1i 11 . 2 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐵) ∈ V)
12 rspe 3149 . . . . . . 7 ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
134, 7elrnmpti 5545 . . . . . . 7 (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
1412, 13sylibr 225 . . . . . 6 ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
1514adantll 705 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
16 simpr 477 . . . . . . 7 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
1716reseq1d 5564 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
18 actfunsn.1 . . . . . . . . . 10 ((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶𝑚 𝐵))
1918sselda 3761 . . . . . . . . 9 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → 𝑧 ∈ (𝐶𝑚 𝐵))
20 elmapfn 8083 . . . . . . . . 9 (𝑧 ∈ (𝐶𝑚 𝐵) → 𝑧 Fn 𝐵)
2119, 20syl 17 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → 𝑧 Fn 𝐵)
22 actfunsn.3 . . . . . . . . . 10 (𝜑𝐼𝑉)
23 fnsng 6119 . . . . . . . . . 10 ((𝐼𝑉𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
2422, 23sylan 575 . . . . . . . . 9 ((𝜑𝑘𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
2524adantr 472 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
26 actfunsn.4 . . . . . . . . . . 11 (𝜑 → ¬ 𝐼𝐵)
27 disjsn 4402 . . . . . . . . . . 11 ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼𝐵)
2826, 27sylibr 225 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
2928adantr 472 . . . . . . . . 9 ((𝜑𝑘𝐶) → (𝐵 ∩ {𝐼}) = ∅)
3029adantr 472 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → (𝐵 ∩ {𝐼}) = ∅)
31 fnunres1 29865 . . . . . . . 8 ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ (𝐵 ∩ {𝐼}) = ∅) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3221, 25, 30, 31syl3anc 1490 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑧𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3332adantr 472 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
3417, 33eqtr2d 2800 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 = (𝑦𝐵))
3515, 34jca 507 . . . 4 ((((𝜑𝑘𝐶) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵)))
3635anasss 458 . . 3 (((𝜑𝑘𝐶) ∧ (𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))) → (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵)))
37 simpr 477 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑧 = (𝑦𝐵))
38 simpr 477 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3938reseq1d 5564 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
4018ad3antrrr 721 . . . . . . . . . . . . 13 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐴 ⊆ (𝐶𝑚 𝐵))
41 simplr 785 . . . . . . . . . . . . 13 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧𝐴)
4240, 41sseldd 3762 . . . . . . . . . . . 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 724 . . . . . . . . . . . 12 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐼𝑉)
45 simp-4r 803 . . . . . . . . . . . 12 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑘𝐶)
4644, 45, 23syl2anc 579 . . . . . . . . . . 11 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
4728ad4antr 724 . . . . . . . . . . 11 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝐵 ∩ {𝐼}) = ∅)
4843, 46, 47, 31syl3anc 1490 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
4948, 41eqeltrd 2844 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) ∈ 𝐴)
5039, 49eqeltrd 2844 . . . . . . . 8 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) ∈ 𝐴)
51 simpr 477 . . . . . . . . 9 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
5251, 13sylib 209 . . . . . . . 8 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑧𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5350, 52r19.29a 3225 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐵) ∈ 𝐴)
5453adantr 472 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑦𝐵) ∈ 𝐴)
5537, 54eqeltrd 2844 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑧𝐴)
5637uneq1d 3928 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) = ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}))
5739, 48eqtrd 2799 . . . . . . . . . 10 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦𝐵) = 𝑧)
5857uneq1d 3928 . . . . . . . . 9 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5958, 38eqtr4d 2802 . . . . . . . 8 (((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6059, 52r19.29a 3225 . . . . . . 7 (((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6160adantr 472 . . . . . 6 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → ((𝑦𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
6256, 61eqtr2d 2800 . . . . 5 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
6355, 62jca 507 . . . 4 ((((𝜑𝑘𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦𝐵)) → (𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
6463anasss 458 . . 3 (((𝜑𝑘𝐶) ∧ (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵))) → (𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
6536, 64impbida 835 . 2 ((𝜑𝑘𝐶) → ((𝑧𝐴𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) ↔ (𝑦 ∈ ran 𝐹𝑧 = (𝑦𝐵))))
664, 8, 11, 65f1od 7083 1 ((𝜑𝑘𝐶) → 𝐹:𝐴1-1-onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wcel 2155  wrex 3056  Vcvv 3350  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334  cop 4340  cmpt 4888  ran crn 5278  cres 5279   Fn wfn 6063  1-1-ontowf1o 6067  (class class class)co 6842  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062
This theorem is referenced by:  breprexplema  31159
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