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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 34869. 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 3852 | . . 3 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ [𝐹 / 𝑓][1o / 𝑛]𝜑) |
4 | bnj125.1 | . . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
5 | bnj105 34717 | . . . . . 6 ⊢ 1o ∈ V | |
6 | 4, 5 | bnj91 34854 | . . . . 5 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
7 | 6 | sbcbii 3852 | . . . 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 34857 | . . . . 5 ⊢ 𝐹 ∈ V |
10 | fveq1 6906 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅)) | |
11 | 10 | eqeq1d 2737 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))) |
12 | 9, 11 | sbcie 3835 | . . . 4 ⊢ ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
13 | 7, 12 | bitri 275 | . . 3 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
14 | 3, 13 | bitri 275 | . 2 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
15 | 1, 14 | bitri 275 | 1 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 [wsbc 3791 ∅c0 4339 {csn 4631 〈cop 4637 ‘cfv 6563 1oc1o 8498 predc-bnj14 34681 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-br 5149 df-suc 6392 df-iota 6516 df-fv 6571 df-1o 8505 |
This theorem is referenced by: bnj150 34869 bnj153 34873 |
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