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Theorem setis 44728
Description: Version of setrec2 44726 expressed as an induction schema. This theorem is a generalization of tfis3 7561. (Contributed by Emmett Weisz, 27-Feb-2022.)
Hypotheses
Ref Expression
setis.1 𝐵 = setrecs(𝐹)
setis.2 (𝑏 = 𝐴 → (𝜓𝜒))
setis.3 (𝜑 → ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
Assertion
Ref Expression
setis (𝜑 → (𝐴𝐵𝜒))
Distinct variable groups:   𝐹,𝑎,𝑏   𝜓,𝑎   𝜒,𝑏   𝐴,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝜓(𝑏)   𝜒(𝑎)   𝐴(𝑎)   𝐵(𝑎,𝑏)

Proof of Theorem setis
StepHypRef Expression
1 setis.1 . . . 4 𝐵 = setrecs(𝐹)
2 setis.3 . . . . 5 (𝜑 → ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
3 ssabral 4039 . . . . . . 7 (𝑎 ⊆ {𝑏𝜓} ↔ ∀𝑏𝑎 𝜓)
4 ssabral 4039 . . . . . . 7 ((𝐹𝑎) ⊆ {𝑏𝜓} ↔ ∀𝑏 ∈ (𝐹𝑎)𝜓)
53, 4imbi12i 352 . . . . . 6 ((𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ (∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
65albii 1811 . . . . 5 (∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
72, 6sylibr 235 . . . 4 (𝜑 → ∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}))
81, 7setrec2v 44727 . . 3 (𝜑𝐵 ⊆ {𝑏𝜓})
98sseld 3963 . 2 (𝜑 → (𝐴𝐵𝐴 ∈ {𝑏𝜓}))
10 setis.2 . . 3 (𝑏 = 𝐴 → (𝜓𝜒))
1110elabg 3663 . 2 (𝐴𝐵 → (𝐴 ∈ {𝑏𝜓} ↔ 𝜒))
129, 11mpbidi 242 1 (𝜑 → (𝐴𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526   = wceq 1528  wcel 2105  {cab 2796  wral 3135  wss 3933  cfv 6348  setrecscsetrecs 44714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-setrecs 44715
This theorem is referenced by: (None)
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