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Theorem setis 48129
Description: Version of setrec2 48126 expressed as an induction schema. This theorem is a generalization of tfis3 7862. (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 4057 . . . . . . 7 (𝑎 ⊆ {𝑏𝜓} ↔ ∀𝑏𝑎 𝜓)
4 ssabral 4057 . . . . . . 7 ((𝐹𝑎) ⊆ {𝑏𝜓} ↔ ∀𝑏 ∈ (𝐹𝑎)𝜓)
53, 4imbi12i 350 . . . . . 6 ((𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ (∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
65albii 1814 . . . . 5 (∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
72, 6sylibr 233 . . . 4 (𝜑 → ∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}))
81, 7setrec2v 48127 . . 3 (𝜑𝐵 ⊆ {𝑏𝜓})
98sseld 3979 . 2 (𝜑 → (𝐴𝐵𝐴 ∈ {𝑏𝜓}))
10 setis.2 . . 3 (𝑏 = 𝐴 → (𝜓𝜒))
1110elabg 3665 . 2 (𝐴𝐵 → (𝐴 ∈ {𝑏𝜓} ↔ 𝜒))
129, 11mpbidi 240 1 (𝜑 → (𝐴𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532   = wceq 1534  wcel 2099  {cab 2705  wral 3058  wss 3947  cfv 6548  setrecscsetrecs 48114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-setrecs 48115
This theorem is referenced by:  pgindnf  48147
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