Theorem bnj958 35637

Description: Technical lemma for bnj69 35707. 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
bnj958.1	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj958.2	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})

Assertion

Ref	Expression
bnj958	⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))

Distinct variable groups:   𝑦,𝑓   𝑦,𝑖   𝑦,𝑛

Allowed substitution hints:   𝐴(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝐶(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝑅(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝐺(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)

Proof of Theorem bnj958

Step	Hyp	Ref	Expression
1		bnj958.2	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
2		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑦𝑓
3		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑦𝑛
4		bnj958.1	. . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
5		nfiu1 4989	. . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
6	4, 5	nfcxfr 2924	. . . . . . . 8 ⊢ Ⅎ𝑦𝐶
7	3, 6	nfop 4852	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝑛, 𝐶⟩
8	7	nfsn 4668	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑛, 𝐶⟩}
9	2, 8	nfun 4172	. . . . 5 ⊢ Ⅎ𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
10	1, 9	nfcxfr 2924	. . . 4 ⊢ Ⅎ𝑦𝐺
11		nfcv 2926	. . . 4 ⊢ Ⅎ𝑦𝑖
12	10, 11	nffv 6915	. . 3 ⊢ Ⅎ𝑦(𝐺‘𝑖)
13	12	nfeq1 2941	. 2 ⊢ Ⅎ𝑦(𝐺‘𝑖) = (𝑓‘𝑖)
14	13	nf5ri 2235	1 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570   ∪ cun 4152  {csn 4588  ⟨cop 4838  ∪ ciun 4954  ‘cfv 6555   predc-bnj14 35386

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-iota 6509  df-fv 6563

This theorem is used by:  bnj966  35641  bnj967  35642