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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1529 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1522 33912. 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 1917 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) | |
3 | bnj1529.2 | . . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) | |
4 | 3 | nfcii 2886 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
5 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6888 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑥𝐺 | |
8 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑥 pred(𝑦, 𝐴, 𝑅) | |
9 | 4, 8 | nfres 5975 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅)) |
10 | 5, 9 | nfop 4882 | . . . . 5 ⊢ Ⅎ𝑥〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉 |
11 | 7, 10 | nffv 6888 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
12 | 6, 11 | nfeq 2915 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
13 | fveq2 6878 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
14 | id 22 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
15 | bnj602 33755 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅)) | |
16 | 15 | reseq2d 5973 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))) |
17 | 14, 16 | opeq12d 4874 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
18 | 17 | fveq2d 6882 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
19 | 13, 18 | eqeq12d 2747 | . . 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 3302 | . 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 217 | 1 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∀wral 3060 〈cop 4628 ↾ cres 5671 ‘cfv 6532 predc-bnj14 33528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-xp 5675 df-res 5681 df-iota 6484 df-fv 6540 df-bnj14 33529 |
This theorem is referenced by: bnj1523 33911 |
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