Theorem bnj958 32207

Description: Technical lemma for bnj69 32277. 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 2977	. . . . . 6 ⊢ Ⅎ𝑦𝑓
3		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑦𝑛
4		bnj958.1	. . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
5		nfiu1 4945	. . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
6	4, 5	nfcxfr 2975	. . . . . . . 8 ⊢ Ⅎ𝑦𝐶
7	3, 6	nfop 4812	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝑛, 𝐶⟩
8	7	nfsn 4636	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑛, 𝐶⟩}
9	2, 8	nfun 4140	. . . . 5 ⊢ Ⅎ𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
10	1, 9	nfcxfr 2975	. . . 4 ⊢ Ⅎ𝑦𝐺
11		nfcv 2977	. . . 4 ⊢ Ⅎ𝑦𝑖
12	10, 11	nffv 6674	. . 3 ⊢ Ⅎ𝑦(𝐺‘𝑖)
13	12	nfeq1 2993	. 2 ⊢ Ⅎ𝑦(𝐺‘𝑖) = (𝑓‘𝑖)
14	13	nf5ri 2191	1 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))

Syntax hints:   → wi 4  ∀wal 1531   = wceq 1533   ∪ cun 3933  {csn 4560  ⟨cop 4566  ∪ ciun 4911  ‘cfv 6349   predc-bnj14 31953

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-iota 6308  df-fv 6357

This theorem is referenced by:  bnj966  32211  bnj967  32212