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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1529 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1522 30394. 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 1830 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) | |
3 | bnj1529.2 | . . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) | |
4 | 3 | nfcii 2742 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
5 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6110 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝐺 | |
8 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥 pred(𝑦, 𝐴, 𝑅) | |
9 | 4, 8 | nfres 5319 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅)) |
10 | 5, 9 | nfop 4356 | . . . . 5 ⊢ Ⅎ𝑥〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉 |
11 | 7, 10 | nffv 6110 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
12 | 6, 11 | nfeq 2762 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
13 | fveq2 6103 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
14 | id 22 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
15 | bnj602 30239 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅)) | |
16 | 15 | reseq2d 5317 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))) |
17 | 14, 16 | opeq12d 4348 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
18 | 17 | fveq2d 6107 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
19 | 13, 18 | eqeq12d 2625 | . . 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 | cbvral 3143 | . 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 207 | 1 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∀wral 2896 〈cop 4131 ↾ cres 5040 ‘cfv 5804 predc-bnj14 30007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-res 5050 df-iota 5768 df-fv 5812 df-bnj14 30008 |
This theorem is referenced by: bnj1523 30393 |
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