Theorem nregmodelf1o 44989

Description: Define a permutation 𝐹 used to produce a model in which ax-reg 9503 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 6807	. . 3 ⊢ I :V–1-1-onto→V
2		0ex 5249	. . 3 ⊢ ∅ ∈ V
3		snex 5378	. . 3 ⊢ {∅} ∈ V
4		f1ofvswap 7247	. . 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 6903	. . . . . . . 8 ⊢ ({∅} ∈ V → ( I ‘{∅}) = {∅})
8	3, 7	ax-mp 5	. . . . . . 7 ⊢ ( I ‘{∅}) = {∅}
9	8	opeq2i 4831	. . . . . 6 ⊢ ⟨∅, ( I ‘{∅})⟩ = ⟨∅, {∅}⟩
10		fvi 6903	. . . . . . . 8 ⊢ (∅ ∈ V → ( I ‘∅) = ∅)
11	2, 10	ax-mp 5	. . . . . . 7 ⊢ ( I ‘∅) = ∅
12	11	opeq2i 4831	. . . . . 6 ⊢ ⟨{∅}, ( I ‘∅)⟩ = ⟨{∅}, ∅⟩
13	9, 12	preq12i 4692	. . . . 5 ⊢ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩} = {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩}
14	13	uneq2i 4118	. . . 4 ⊢ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}) = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
15	6, 14	eqtr4i 2755	. . 3 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩})
16		f1oeq1 6756	. . 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 1540   ∈ wcel 2109  Vcvv 3438   ∖ cdif 3902   ∪ cun 3903  ∅c0 4286  {csn 4579  {cpr 4581  ⟨cop 4585   I cid 5517   ↾ cres 5625  –1-1-onto→wf1o 6485  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494

This theorem is referenced by:  nregmodellem  44990  nregmodelaxext  44992