Theorem bnj958 34933

Description: Technical lemma for bnj69 35003. 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 2903	. . . . . 6 ⊢ Ⅎ𝑦𝑓
3		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑦𝑛
4		bnj958.1	. . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
5		nfiu1 5032	. . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
6	4, 5	nfcxfr 2901	. . . . . . . 8 ⊢ Ⅎ𝑦𝐶
7	3, 6	nfop 4894	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝑛, 𝐶⟩
8	7	nfsn 4712	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑛, 𝐶⟩}
9	2, 8	nfun 4180	. . . . 5 ⊢ Ⅎ𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
10	1, 9	nfcxfr 2901	. . . 4 ⊢ Ⅎ𝑦𝐺
11		nfcv 2903	. . . 4 ⊢ Ⅎ𝑦𝑖
12	10, 11	nffv 6917	. . 3 ⊢ Ⅎ𝑦(𝐺‘𝑖)
13	12	nfeq1 2919	. 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 1535   = wceq 1537   ∪ cun 3961  {csn 4631  ⟨cop 4637  ∪ ciun 4996  ‘cfv 6563   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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-iota 6516  df-fv 6571

This theorem is referenced by:  bnj966  34937  bnj967  34938