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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj125 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 31463. 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 |
---|---|
bnj125.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
bnj125.2 | ⊢ (𝜑′ ↔ [1𝑜 / 𝑛]𝜑) |
bnj125.3 | ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′) |
bnj125.4 | ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
Ref | Expression |
---|---|
bnj125 | ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj125.3 | . 2 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′) | |
2 | bnj125.2 | . . . 4 ⊢ (𝜑′ ↔ [1𝑜 / 𝑛]𝜑) | |
3 | 2 | sbcbii 3689 | . . 3 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ [𝐹 / 𝑓][1𝑜 / 𝑛]𝜑) |
4 | bnj125.1 | . . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
5 | bnj105 31310 | . . . . . 6 ⊢ 1𝑜 ∈ V | |
6 | 4, 5 | bnj91 31448 | . . . . 5 ⊢ ([1𝑜 / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
7 | 6 | sbcbii 3689 | . . . 4 ⊢ ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜑 ↔ [𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
8 | bnj125.4 | . . . . . 6 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
9 | 8 | bnj95 31451 | . . . . 5 ⊢ 𝐹 ∈ V |
10 | fveq1 6410 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅)) | |
11 | 10 | eqeq1d 2801 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))) |
12 | 9, 11 | sbcie 3668 | . . . 4 ⊢ ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
13 | 7, 12 | bitri 267 | . . 3 ⊢ ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
14 | 3, 13 | bitri 267 | . 2 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
15 | 1, 14 | bitri 267 | 1 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1653 [wsbc 3633 ∅c0 4115 {csn 4368 〈cop 4374 ‘cfv 6101 1𝑜c1o 7792 predc-bnj14 31274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rex 3095 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-pw 4351 df-sn 4369 df-pr 4371 df-uni 4629 df-br 4844 df-suc 5947 df-iota 6064 df-fv 6109 df-1o 7799 |
This theorem is referenced by: bnj150 31463 bnj153 31467 |
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