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Theorem setis 45166
 Description: Version of setrec2 45164 expressed as an induction schema. This theorem is a generalization of tfis3 7557. (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 4017 . . . . . . 7 (𝑎 ⊆ {𝑏𝜓} ↔ ∀𝑏𝑎 𝜓)
4 ssabral 4017 . . . . . . 7 ((𝐹𝑎) ⊆ {𝑏𝜓} ↔ ∀𝑏 ∈ (𝐹𝑎)𝜓)
53, 4imbi12i 354 . . . . . 6 ((𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ (∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
65albii 1821 . . . . 5 (∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}) ↔ ∀𝑎(∀𝑏𝑎 𝜓 → ∀𝑏 ∈ (𝐹𝑎)𝜓))
72, 6sylibr 237 . . . 4 (𝜑 → ∀𝑎(𝑎 ⊆ {𝑏𝜓} → (𝐹𝑎) ⊆ {𝑏𝜓}))
81, 7setrec2v 45165 . . 3 (𝜑𝐵 ⊆ {𝑏𝜓})
98sseld 3941 . 2 (𝜑 → (𝐴𝐵𝐴 ∈ {𝑏𝜓}))
10 setis.2 . . 3 (𝑏 = 𝐴 → (𝜓𝜒))
1110elabg 3641 . 2 (𝐴𝐵 → (𝐴 ∈ {𝑏𝜓} ↔ 𝜒))
129, 11mpbidi 244 1 (𝜑 → (𝐴𝐵𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2114  {cab 2800  ∀wral 3130   ⊆ wss 3908  ‘cfv 6334  setrecscsetrecs 45152 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342  df-setrecs 45153 This theorem is referenced by: (None)
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