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Theorem setis 47990
Description: Version of setrec2 47987 expressed as an induction schema. This theorem is a generalization of tfis3 7841. (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 4052 . . . . . . 7 (𝑎 ⊆ {𝑏𝜓} ↔ ∀𝑏𝑎 𝜓)
4 ssabral 4052 . . . . . . 7 ((𝐹𝑎) ⊆ {𝑏𝜓} ↔ ∀𝑏 ∈ (𝐹𝑎)𝜓)
53, 4imbi12i 350 . . . . . 6 ((𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ (∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
65albii 1813 . . . . 5 (∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
72, 6sylibr 233 . . . 4 (𝜑 → ∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}))
81, 7setrec2v 47988 . . 3 (𝜑𝐵 ⊆ {𝑏𝜓})
98sseld 3974 . 2 (𝜑 → (𝐴𝐵𝐴 ∈ {𝑏𝜓}))
10 setis.2 . . 3 (𝑏 = 𝐴 → (𝜓𝜒))
1110elabg 3659 . 2 (𝐴𝐵 → (𝐴 ∈ {𝑏𝜓} ↔ 𝜒))
129, 11mpbidi 240 1 (𝜑 → (𝐴𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  wcel 2098  {cab 2701  wral 3053  wss 3941  cfv 6534  setrecscsetrecs 47975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fv 6542  df-setrecs 47976
This theorem is referenced by:  pgindnf  48008
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