Theorem bnj125 35368

Description: Technical lemma for bnj150 35372. 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

Step	Hyp	Ref	Expression
1		bnj125.3	. 2 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
2		bnj125.2	. . . 4 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
3	2	sbcbii 3798	. . 3 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ [𝐹 / 𝑓][1o / 𝑛]𝜑)
4		bnj125.1	. . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5		bnj105 35221	. . . . . 6 ⊢ 1o ∈ V
6	4, 5	bnj91 35357	. . . . 5 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
7	6	sbcbii 3798	. . . 4 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ [𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8		bnj125.4	. . . . . 6 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
9	8	bnj95 35360	. . . . 5 ⊢ 𝐹 ∈ V
10		fveq1 6881	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅))
11	10	eqeq1d 2764	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)))
12	9, 11	sbcie 3783	. . . 4 ⊢ ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
13	7, 12	bitri 278	. . 3 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
14	3, 13	bitri 278	. 2 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
15	1, 14	bitri 278	1 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  [wsbc 3742  ∅c0 4282  {csn 4587  ⟨cop 4593  ‘cfv 6537  1_oc1o 8451   predc-bnj14 35185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-pw 4562  df-sn 4588  df-pr 4590  df-uni 4871  df-br 5108  df-suc 6367  df-iota 6493  df-fv 6545  df-1o 8458

This theorem is used by:  bnj150  35372  bnj153  35376