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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1529 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1522 32241. 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 |
---|---|
bnj1529.1 | ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
bnj1529.2 | ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) |
Ref | Expression |
---|---|
bnj1529 | ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1529.1 | . 2 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) | |
2 | nfv 1906 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) | |
3 | bnj1529.2 | . . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) | |
4 | 3 | nfcii 2962 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
5 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6673 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝐺 | |
8 | nfcv 2974 | . . . . . . 7 ⊢ Ⅎ𝑥 pred(𝑦, 𝐴, 𝑅) | |
9 | 4, 8 | nfres 5848 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅)) |
10 | 5, 9 | nfop 4811 | . . . . 5 ⊢ Ⅎ𝑥〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉 |
11 | 7, 10 | nffv 6673 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
12 | 6, 11 | nfeq 2988 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
13 | fveq2 6663 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
14 | id 22 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
15 | bnj602 32086 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅)) | |
16 | 15 | reseq2d 5846 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))) |
17 | 14, 16 | opeq12d 4803 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
18 | 17 | fveq2d 6667 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
19 | 13, 18 | eqeq12d 2834 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) ↔ (𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉))) |
20 | 2, 12, 19 | cbvralw 3439 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) ↔ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
21 | 1, 20 | sylib 219 | 1 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 = wceq 1528 ∈ wcel 2105 ∀wral 3135 〈cop 4563 ↾ cres 5550 ‘cfv 6348 predc-bnj14 31857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-res 5560 df-iota 6307 df-fv 6356 df-bnj14 31858 |
This theorem is referenced by: bnj1523 32240 |
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