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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj125 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 30946. 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 3491 | . . 3 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ [𝐹 / 𝑓][1𝑜 / 𝑛]𝜑) |
4 | bnj125.1 | . . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
5 | bnj105 30790 | . . . . . 6 ⊢ 1𝑜 ∈ V | |
6 | 4, 5 | bnj91 30931 | . . . . 5 ⊢ ([1𝑜 / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
7 | 6 | sbcbii 3491 | . . . 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 30934 | . . . . 5 ⊢ 𝐹 ∈ V |
10 | fveq1 6190 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅)) | |
11 | 10 | eqeq1d 2624 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))) |
12 | 9, 11 | sbcie 3470 | . . . 4 ⊢ ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
13 | 7, 12 | bitri 264 | . . 3 ⊢ ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
14 | 3, 13 | bitri 264 | . 2 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
15 | 1, 14 | bitri 264 | 1 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 [wsbc 3435 ∅c0 3915 {csn 4177 〈cop 4183 ‘cfv 5888 1𝑜c1o 7553 predc-bnj14 30754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 df-uni 4437 df-br 4654 df-suc 5729 df-iota 5851 df-fv 5896 df-1o 7560 |
This theorem is referenced by: bnj150 30946 bnj153 30950 |
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