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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 34638. 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 3834 | . . 3 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ [𝐹 / 𝑓][1o / 𝑛]𝜑) |
4 | bnj125.1 | . . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
5 | bnj105 34486 | . . . . . 6 ⊢ 1o ∈ V | |
6 | 4, 5 | bnj91 34623 | . . . . 5 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
7 | 6 | sbcbii 3834 | . . . 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 34626 | . . . . 5 ⊢ 𝐹 ∈ V |
10 | fveq1 6895 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅)) | |
11 | 10 | eqeq1d 2727 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))) |
12 | 9, 11 | sbcie 3817 | . . . 4 ⊢ ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
13 | 7, 12 | bitri 274 | . . 3 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
14 | 3, 13 | bitri 274 | . 2 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
15 | 1, 14 | bitri 274 | 1 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 [wsbc 3773 ∅c0 4322 {csn 4630 〈cop 4636 ‘cfv 6549 1oc1o 8480 predc-bnj14 34450 |
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-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-pw 4606 df-sn 4631 df-pr 4633 df-uni 4910 df-br 5150 df-suc 6377 df-iota 6501 df-fv 6557 df-1o 8487 |
This theorem is referenced by: bnj150 34638 bnj153 34642 |
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