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Theorem nregmodelf1o 45694
Description: Define a permutation 𝐹 used to produce a model in which ax-reg 9553 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 6861 . . 3 I :V–1-1-onto→V
2 0ex 5269 . . 3 ∅ ∈ V
3 snex 5410 . . 3 {∅} ∈ V
4 f1ofvswap 7304 . . 3 (( I :V–1-1-onto→V ∧ ∅ ∈ V ∧ {∅} ∈ V) → (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V)
51, 2, 3, 4mp3an 1488 . 2 (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V
6 nregmodel.1 . . . 4 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
7 fvi 6957 . . . . . . . 8 ({∅} ∈ V → ( I ‘{∅}) = {∅})
83, 7ax-mp 5 . . . . . . 7 ( I ‘{∅}) = {∅}
98opeq2i 4841 . . . . . 6 ⟨∅, ( I ‘{∅})⟩ = ⟨∅, {∅}⟩
10 fvi 6957 . . . . . . . 8 (∅ ∈ V → ( I ‘∅) = ∅)
112, 10ax-mp 5 . . . . . . 7 ( I ‘∅) = ∅
1211opeq2i 4841 . . . . . 6 ⟨{∅}, ( I ‘∅)⟩ = ⟨{∅}, ∅⟩
139, 12preq12i 4703 . . . . 5 {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩} = {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩}
1413uneq2i 4118 . . . 4 (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}) = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
156, 14eqtr4i 2787 . . 3 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩})
16 f1oeq1 6808 . . 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 1568  wcel 2141  Vcvv 3453  cdif 3901  cun 3902  c0 4285  {csn 4588  {cpr 4590  cop 4594   I cid 5555  cres 5663  1-1-ontowf1o 6535  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544
This theorem is referenced by:  nregmodellem  45695  nregmodelaxext  45697
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