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

Step	Hyp	Ref	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)
5	1, 2, 3, 4	mp3an 1488	. 2 ⊢ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}):V–1-1-onto→V
6		nregmodel.1	. . . 4 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
7		fvi 6957	. . . . . . . 8 ⊢ ({∅} ∈ V → ( I ‘{∅}) = {∅})
8	3, 7	ax-mp 5	. . . . . . 7 ⊢ ( I ‘{∅}) = {∅}
9	8	opeq2i 4841	. . . . . 6 ⊢ ⟨∅, ( I ‘{∅})⟩ = ⟨∅, {∅}⟩
10		fvi 6957	. . . . . . . 8 ⊢ (∅ ∈ V → ( I ‘∅) = ∅)
11	2, 10	ax-mp 5	. . . . . . 7 ⊢ ( I ‘∅) = ∅
12	11	opeq2i 4841	. . . . . 6 ⊢ ⟨{∅}, ( I ‘∅)⟩ = ⟨{∅}, ∅⟩
13	9, 12	preq12i 4703	. . . . 5 ⊢ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩} = {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩}
14	13	uneq2i 4118	. . . 4 ⊢ (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, ( I ‘{∅})⟩, ⟨{∅}, ( I ‘∅)⟩}) = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
15	6, 14	eqtr4i 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))
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 234	1 ⊢ 𝐹:V–1-1-onto→V

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-onto→wf1o 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