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Theorem permaxsep 45544
Description: The Axiom of Separation ax-sep 5243 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148.

Note that, to prove that an instance of Separation holds in the model, 𝜑 would need have all instances of replaced with 𝑅. But this still results in an instance of this theorem, so we do establish that Separation holds. (Contributed by Eric Schmidt, 6-Nov-2025.)

Hypotheses
Ref Expression
permmodel.1 𝐹:V–1-1-onto→V
permmodel.2 𝑅 = (𝐹 ∘ E )
Assertion
Ref Expression
permaxsep 𝑦𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧   𝑥,𝐹,𝑦   𝑦,𝑅
Allowed substitution hints:   𝜑(𝑥)   𝑅(𝑥,𝑧)   𝐹(𝑧)

Proof of Theorem permaxsep
StepHypRef Expression
1 fvex 6875 . 2 (𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}) ∈ V
2 nfcv 2923 . . . . 5 𝑥𝐹
3 nfrab1 3433 . . . . 5 𝑥{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}
42, 3nffv 6872 . . . 4 𝑥(𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑})
54nfeq2 2940 . . 3 𝑥 𝑦 = (𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑})
6 breq2 5101 . . . 4 (𝑦 = (𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}) → (𝑥𝑅𝑦𝑥𝑅(𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑})))
76bibi1d 345 . . 3 (𝑦 = (𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}) → ((𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧𝜑)) ↔ (𝑥𝑅(𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧𝜑))))
85, 7albid 2256 . 2 (𝑦 = (𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}) → (∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧𝜑)) ↔ ∀𝑥(𝑥𝑅(𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧𝜑))))
9 permmodel.1 . . . . 5 𝐹:V–1-1-onto→V
10 permmodel.2 . . . . 5 𝑅 = (𝐹 ∘ E )
11 vex 3457 . . . . 5 𝑥 ∈ V
12 fvex 6875 . . . . . 6 (𝐹𝑧) ∈ V
1312rabex 5292 . . . . 5 {𝑥 ∈ (𝐹𝑧) ∣ 𝜑} ∈ V
149, 10, 11, 13brpermmodelcnv 45541 . . . 4 (𝑥𝑅(𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐹𝑧) ∣ 𝜑})
15 rabid 3434 . . . . 5 (𝑥 ∈ {𝑥 ∈ (𝐹𝑧) ∣ 𝜑} ↔ (𝑥 ∈ (𝐹𝑧) ∧ 𝜑))
16 vex 3457 . . . . . . 7 𝑧 ∈ V
179, 10, 11, 16brpermmodel 45540 . . . . . 6 (𝑥𝑅𝑧𝑥 ∈ (𝐹𝑧))
1817bicomi 226 . . . . 5 (𝑥 ∈ (𝐹𝑧) ↔ 𝑥𝑅𝑧)
1915, 18bianbi 636 . . . 4 (𝑥 ∈ {𝑥 ∈ (𝐹𝑧) ∣ 𝜑} ↔ (𝑥𝑅𝑧𝜑))
2014, 19bitri 277 . . 3 (𝑥𝑅(𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧𝜑))
2120ax-gen 1814 . 2 𝑥(𝑥𝑅(𝐹‘{𝑥 ∈ (𝐹𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧𝜑))
221, 8, 21ceqsexv2d 3502 1 𝑦𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 399  wal 1557   = wceq 1559  wex 1798  wcel 2141  {crab 3413  Vcvv 3453   class class class wbr 5097   E cep 5542  ccnv 5642  ccom 5647  1-1-ontowf1o 6515  cfv 6516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-eprel 5543  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524
This theorem is referenced by: (None)
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