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Theorem setis 47229
Description: Version of setrec2 47226 expressed as an induction schema. This theorem is a generalization of tfis3 7795. (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 4020 . . . . . . 7 (𝑎 ⊆ {𝑏𝜓} ↔ ∀𝑏𝑎 𝜓)
4 ssabral 4020 . . . . . . 7 ((𝐹𝑎) ⊆ {𝑏𝜓} ↔ ∀𝑏 ∈ (𝐹𝑎)𝜓)
53, 4imbi12i 351 . . . . . 6 ((𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ (∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
65albii 1822 . . . . 5 (∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
72, 6sylibr 233 . . . 4 (𝜑 → ∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}))
81, 7setrec2v 47227 . . 3 (𝜑𝐵 ⊆ {𝑏𝜓})
98sseld 3944 . 2 (𝜑 → (𝐴𝐵𝐴 ∈ {𝑏𝜓}))
10 setis.2 . . 3 (𝑏 = 𝐴 → (𝜓𝜒))
1110elabg 3629 . 2 (𝐴𝐵 → (𝐴 ∈ {𝑏𝜓} ↔ 𝜒))
129, 11mpbidi 240 1 (𝜑 → (𝐴𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wss 3911  cfv 6497  setrecscsetrecs 47214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-setrecs 47215
This theorem is referenced by:  pgindnf  47247
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