Theorem nregmodelf1o 45292

Description: Define a permutation 𝐹 used to produce a model in which ax-reg 9501 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

Step	Hyp	Ref	Expression
1		f1ovi 6815	. . 3 ⊢ I :V–1-1-onto→V
2		0ex 5253	. . 3 ⊢ ∅ ∈ V
3		snex 5382	. . 3 ⊢ {∅} ∈ V
4		f1ofvswap 7254	. . 3 ⊢ (( I :V–1-1-onto→V ∧ ∅ ∈ V ∧ {∅} ∈ V) → (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V)
5	1, 2, 3, 4	mp3an 1464	. 2 ⊢ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V
6		nregmodel.1	. . . 4 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
7		fvi 6911	. . . . . . . 8 ⊢ ({∅} ∈ V → ( I ‘{∅}) = {∅})
8	3, 7	ax-mp 5	. . . . . . 7 ⊢ ( I ‘{∅}) = {∅}
9	8	opeq2i 4834	. . . . . 6 ⊢ ⟨∅, ( I ‘{∅})⟩ = ⟨∅, {∅}⟩
10		fvi 6911	. . . . . . . 8 ⊢ (∅ ∈ V → ( I ‘∅) = ∅)
11	2, 10	ax-mp 5	. . . . . . 7 ⊢ ( I ‘∅) = ∅
12	11	opeq2i 4834	. . . . . 6 ⊢ ⟨{∅}, ( I ‘∅)⟩ = ⟨{∅}, ∅⟩
13	9, 12	preq12i 4696	. . . . 5 ⊢ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩} = {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩}
14	13	uneq2i 4118	. . . 4 ⊢ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}) = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
15	6, 14	eqtr4i 2763	. . 3 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩})
16		f1oeq1 6763	. . 3 ⊢ (𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}) → (𝐹:V–1-1-onto→V ↔ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V))
17	15, 16	ax-mp 5	. 2 ⊢ (𝐹:V–1-1-onto→V ↔ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V)
18	5, 17	mpbir 231	1 ⊢ 𝐹:V–1-1-onto→V

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∖ cdif 3899   ∪ cun 3900  ∅c0 4286  {csn 4581  {cpr 4583  ⟨cop 4587   I cid 5519   ↾ cres 5627  –1-1-onto→wf1o 6492  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501

This theorem is referenced by:  nregmodellem  45293  nregmodelaxext  45295