Theorem bnj958 31010

Description: Technical lemma for bnj69 31078. 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 2764	. . . . . 6 ⊢ Ⅎ𝑦𝑓
3		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑦𝑛
4		bnj958.1	. . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
5		nfiu1 4550	. . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
6	4, 5	nfcxfr 2762	. . . . . . . 8 ⊢ Ⅎ𝑦𝐶
7	3, 6	nfop 4418	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝑛, 𝐶⟩
8	7	nfsn 4242	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑛, 𝐶⟩}
9	2, 8	nfun 3769	. . . . 5 ⊢ Ⅎ𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
10	1, 9	nfcxfr 2762	. . . 4 ⊢ Ⅎ𝑦𝐺
11		nfcv 2764	. . . 4 ⊢ Ⅎ𝑦𝑖
12	10, 11	nffv 6198	. . 3 ⊢ Ⅎ𝑦(𝐺‘𝑖)
13	12	nfeq1 2778	. 2 ⊢ Ⅎ𝑦(𝐺‘𝑖) = (𝑓‘𝑖)
14	13	nf5ri 2065	1 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))

Syntax hints:   → wi 4  ∀wal 1481   = wceq 1483   ∪ cun 3572  {csn 4177  ⟨cop 4183  ∪ ciun 4520  ‘cfv 5888   predc-bnj14 30754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj966  31014  bnj967  31015