Theorem nregmodelf1o 45027

Description: Define a permutation 𝐹 used to produce a model in which ax-reg 9473 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 6798	. . 3 ⊢ I :V–1-1-onto→V
2		0ex 5243	. . 3 ⊢ ∅ ∈ V
3		snex 5372	. . 3 ⊢ {∅} ∈ V
4		f1ofvswap 7235	. . 3 ⊢ (( I :V–1-1-onto→V ∧ ∅ ∈ V ∧ {∅} ∈ V) → (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V)
5	1, 2, 3, 4	mp3an 1463	. 2 ⊢ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V
6		nregmodel.1	. . . 4 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
7		fvi 6893	. . . . . . . 8 ⊢ ({∅} ∈ V → ( I ‘{∅}) = {∅})
8	3, 7	ax-mp 5	. . . . . . 7 ⊢ ( I ‘{∅}) = {∅}
9	8	opeq2i 4827	. . . . . 6 ⊢ ⟨∅, ( I ‘{∅})⟩ = ⟨∅, {∅}⟩
10		fvi 6893	. . . . . . . 8 ⊢ (∅ ∈ V → ( I ‘∅) = ∅)
11	2, 10	ax-mp 5	. . . . . . 7 ⊢ ( I ‘∅) = ∅
12	11	opeq2i 4827	. . . . . 6 ⊢ ⟨{∅}, ( I ‘∅)⟩ = ⟨{∅}, ∅⟩
13	9, 12	preq12i 4689	. . . . 5 ⊢ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩} = {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩}
14	13	uneq2i 4113	. . . 4 ⊢ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}) = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
15	6, 14	eqtr4i 2756	. . 3 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩})
16		f1oeq1 6747	. . 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 1541   ∈ wcel 2110  Vcvv 3434   ∖ cdif 3897   ∪ cun 3898  ∅c0 4281  {csn 4574  {cpr 4576  ⟨cop 4580   I cid 5508   ↾ cres 5616  –1-1-onto→wf1o 6476  ‘cfv 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485

This theorem is referenced by:  nregmodellem  45028  nregmodelaxext  45030