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Theorem bnj125 34865
Description: Technical lemma for bnj150 34869. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj125.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj125.2 (𝜑′[1o / 𝑛]𝜑)
bnj125.3 (𝜑″[𝐹 / 𝑓]𝜑′)
bnj125.4 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
Assertion
Ref Expression
bnj125 (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑛   𝑓,𝐹   𝑅,𝑓,𝑛   𝑥,𝑓,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝐴(𝑥)   𝑅(𝑥)   𝐹(𝑥,𝑛)   𝜑′(𝑥,𝑓,𝑛)   𝜑″(𝑥,𝑓,𝑛)

Proof of Theorem bnj125
StepHypRef Expression
1 bnj125.3 . 2 (𝜑″[𝐹 / 𝑓]𝜑′)
2 bnj125.2 . . . 4 (𝜑′[1o / 𝑛]𝜑)
32sbcbii 3852 . . 3 ([𝐹 / 𝑓]𝜑′[𝐹 / 𝑓][1o / 𝑛]𝜑)
4 bnj125.1 . . . . . 6 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5 bnj105 34717 . . . . . 6 1o ∈ V
64, 5bnj91 34854 . . . . 5 ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
76sbcbii 3852 . . . 4 ([𝐹 / 𝑓][1o / 𝑛]𝜑[𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8 bnj125.4 . . . . . 6 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
98bnj95 34857 . . . . 5 𝐹 ∈ V
10 fveq1 6906 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅))
1110eqeq1d 2737 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)))
129, 11sbcie 3835 . . . 4 ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
137, 12bitri 275 . . 3 ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
143, 13bitri 275 . 2 ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
151, 14bitri 275 1 (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  [wsbc 3791  c0 4339  {csn 4631  cop 4637  cfv 6563  1oc1o 8498   predc-bnj14 34681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-br 5149  df-suc 6392  df-iota 6516  df-fv 6571  df-1o 8505
This theorem is referenced by:  bnj150  34869  bnj153  34873
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