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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj958 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 34641. 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.) |
Ref | Expression |
---|---|
bnj958.1 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) |
bnj958.2 | ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩}) |
Ref | Expression |
---|---|
bnj958 | ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj958.2 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩}) | |
2 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑦𝑓 | |
3 | nfcv 2899 | . . . . . . . 8 ⊢ Ⅎ𝑦𝑛 | |
4 | bnj958.1 | . . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
5 | nfiu1 5030 | . . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
6 | 4, 5 | nfcxfr 2897 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐶 |
7 | 3, 6 | nfop 4890 | . . . . . . 7 ⊢ Ⅎ𝑦⟨𝑛, 𝐶⟩ |
8 | 7 | nfsn 4712 | . . . . . 6 ⊢ Ⅎ𝑦{⟨𝑛, 𝐶⟩} |
9 | 2, 8 | nfun 4164 | . . . . 5 ⊢ Ⅎ𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩}) |
10 | 1, 9 | nfcxfr 2897 | . . . 4 ⊢ Ⅎ𝑦𝐺 |
11 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑦𝑖 | |
12 | 10, 11 | nffv 6907 | . . 3 ⊢ Ⅎ𝑦(𝐺‘𝑖) |
13 | 12 | nfeq1 2915 | . 2 ⊢ Ⅎ𝑦(𝐺‘𝑖) = (𝑓‘𝑖) |
14 | 13 | nf5ri 2184 | 1 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 = wceq 1534 ∪ cun 3945 {csn 4629 ⟨cop 4635 ∪ ciun 4996 ‘cfv 6548 predc-bnj14 34319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-iota 6500 df-fv 6556 |
This theorem is referenced by: bnj966 34575 bnj967 34576 |
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