Theorem bnj958 32322

Description: Technical lemma for bnj69 32392. 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 2955	. . . . . 6 ⊢ Ⅎ𝑦𝑓
3		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑦𝑛
4		bnj958.1	. . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
5		nfiu1 4915	. . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
6	4, 5	nfcxfr 2953	. . . . . . . 8 ⊢ Ⅎ𝑦𝐶
7	3, 6	nfop 4781	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝑛, 𝐶⟩
8	7	nfsn 4603	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑛, 𝐶⟩}
9	2, 8	nfun 4092	. . . . 5 ⊢ Ⅎ𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
10	1, 9	nfcxfr 2953	. . . 4 ⊢ Ⅎ𝑦𝐺
11		nfcv 2955	. . . 4 ⊢ Ⅎ𝑦𝑖
12	10, 11	nffv 6655	. . 3 ⊢ Ⅎ𝑦(𝐺‘𝑖)
13	12	nfeq1 2970	. 2 ⊢ Ⅎ𝑦(𝐺‘𝑖) = (𝑓‘𝑖)
14	13	nf5ri 2193	1 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))

Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538   ∪ cun 3879  {csn 4525  ⟨cop 4531  ∪ ciun 4881  ‘cfv 6324   predc-bnj14 32068

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

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-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-iota 6283  df-fv 6332

This theorem is referenced by:  bnj966  32326  bnj967  32327