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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 34540. 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 2895 | . . . . . 6 ⊢ Ⅎ𝑦𝑓 | |
3 | nfcv 2895 | . . . . . . . 8 ⊢ Ⅎ𝑦𝑛 | |
4 | bnj958.1 | . . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
5 | nfiu1 5022 | . . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
6 | 4, 5 | nfcxfr 2893 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐶 |
7 | 3, 6 | nfop 4882 | . . . . . . 7 ⊢ Ⅎ𝑦⟨𝑛, 𝐶⟩ |
8 | 7 | nfsn 4704 | . . . . . 6 ⊢ Ⅎ𝑦{⟨𝑛, 𝐶⟩} |
9 | 2, 8 | nfun 4158 | . . . . 5 ⊢ Ⅎ𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩}) |
10 | 1, 9 | nfcxfr 2893 | . . . 4 ⊢ Ⅎ𝑦𝐺 |
11 | nfcv 2895 | . . . 4 ⊢ Ⅎ𝑦𝑖 | |
12 | 10, 11 | nffv 6892 | . . 3 ⊢ Ⅎ𝑦(𝐺‘𝑖) |
13 | 12 | nfeq1 2910 | . 2 ⊢ Ⅎ𝑦(𝐺‘𝑖) = (𝑓‘𝑖) |
14 | 13 | nf5ri 2180 | 1 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ∪ cun 3939 {csn 4621 ⟨cop 4627 ∪ ciun 4988 ‘cfv 6534 predc-bnj14 34218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-iota 6486 df-fv 6542 |
This theorem is referenced by: bnj966 34474 bnj967 34475 |
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