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Theorem nregmodelf1o 45609
Description: Define a permutation 𝐹 used to produce a model in which ax-reg 9550 is false. The permutation swaps and {∅} and leaves the rest of 𝑉 fixed. This is an example given after Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 16-Nov-2025.)
Hypothesis
Ref Expression
nregmodel.1 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
Assertion
Ref Expression
nregmodelf1o 𝐹:V–1-1-onto→V

Proof of Theorem nregmodelf1o
StepHypRef Expression
1 f1ovi 6859 . . 3 I :V–1-1-onto→V
2 0ex 5269 . . 3 ∅ ∈ V
3 snex 5408 . . 3 {∅} ∈ V
4 f1ofvswap 7302 . . 3 (( I :V–1-1-onto→V ∧ ∅ ∈ V ∧ {∅} ∈ V) → (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V)
51, 2, 3, 4mp3an 1487 . 2 (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V
6 nregmodel.1 . . . 4 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
7 fvi 6955 . . . . . . . 8 ({∅} ∈ V → ( I ‘{∅}) = {∅})
83, 7ax-mp 5 . . . . . . 7 ( I ‘{∅}) = {∅}
98opeq2i 4843 . . . . . 6 ⟨∅, ( I ‘{∅})⟩ = ⟨∅, {∅}⟩
10 fvi 6955 . . . . . . . 8 (∅ ∈ V → ( I ‘∅) = ∅)
112, 10ax-mp 5 . . . . . . 7 ( I ‘∅) = ∅
1211opeq2i 4843 . . . . . 6 ⟨{∅}, ( I ‘∅)⟩ = ⟨{∅}, ∅⟩
139, 12preq12i 4706 . . . . 5 {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩} = {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩}
1413uneq2i 4127 . . . 4 (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}) = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
156, 14eqtr4i 2795 . . 3 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩})
16 f1oeq1 6806 . . 3 (𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}) → (𝐹:V–1-1-onto→V ↔ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V))
1715, 16ax-mp 5 . 2 (𝐹:V–1-1-onto→V ↔ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V)
185, 17mpbir 234 1 𝐹:V–1-1-onto→V
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1567  wcel 2149  Vcvv 3463  cdif 3910  cun 3911  c0 4294  {csn 4591  {cpr 4593  cop 4597   I cid 5553  cres 5661  1-1-ontowf1o 6532  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541
This theorem is referenced by:  nregmodellem  45610  nregmodelaxext  45612
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