Theorem nregmodelf1o 45005

Description: Define a permutation 𝐹 used to produce a model in which ax-reg 9545 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 6839	. . 3 ⊢ I :V–1-1-onto→V
2		0ex 5262	. . 3 ⊢ ∅ ∈ V
3		snex 5391	. . 3 ⊢ {∅} ∈ V
4		f1ofvswap 7281	. . 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 6937	. . . . . . . 8 ⊢ ({∅} ∈ V → ( I ‘{∅}) = {∅})
8	3, 7	ax-mp 5	. . . . . . 7 ⊢ ( I ‘{∅}) = {∅}
9	8	opeq2i 4841	. . . . . 6 ⊢ ⟨∅, ( I ‘{∅})⟩ = ⟨∅, {∅}⟩
10		fvi 6937	. . . . . . . 8 ⊢ (∅ ∈ V → ( I ‘∅) = ∅)
11	2, 10	ax-mp 5	. . . . . . 7 ⊢ ( I ‘∅) = ∅
12	11	opeq2i 4841	. . . . . 6 ⊢ ⟨{∅}, ( I ‘∅)⟩ = ⟨{∅}, ∅⟩
13	9, 12	preq12i 4702	. . . . 5 ⊢ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩} = {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩}
14	13	uneq2i 4128	. . . 4 ⊢ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}) = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
15	6, 14	eqtr4i 2755	. . 3 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩})
16		f1oeq1 6788	. . 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 3447   ∖ cdif 3911   ∪ cun 3912  ∅c0 4296  {csn 4589  {cpr 4591  ⟨cop 4595   I cid 5532   ↾ cres 5640  –1-1-onto→wf1o 6510  ‘cfv 6511

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 5251  ax-nul 5261  ax-pr 5387

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519

This theorem is referenced by:  nregmodellem  45006  nregmodelaxext  45008