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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1529 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj1522 35086. 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 1914 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) | |
| 3 | bnj1529.2 | . . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) | |
| 4 | 3 | nfcii 2894 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
| 5 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
| 6 | 4, 5 | nffv 6916 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
| 7 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝐺 | |
| 8 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑥 pred(𝑦, 𝐴, 𝑅) | |
| 9 | 4, 8 | nfres 5999 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ pred(𝑦, 𝐴, 𝑅)) |
| 10 | 5, 9 | nfop 4889 | . . . . 5 ⊢ Ⅎ𝑥〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉 |
| 11 | 7, 10 | nffv 6916 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
| 12 | 6, 11 | nfeq 2919 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
| 13 | fveq2 6906 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
| 14 | id 22 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
| 15 | bnj602 34929 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → pred(𝑥, 𝐴, 𝑅) = pred(𝑦, 𝐴, 𝑅)) | |
| 16 | 15 | reseq2d 5997 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))) |
| 17 | 14, 16 | opeq12d 4881 | . . . . 5 ⊢ (𝑥 = 𝑦 → 〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 = 〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉) |
| 18 | 17 | fveq2d 6910 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
| 19 | 13, 18 | eqeq12d 2753 | . . 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 3306 | . 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 218 | 1 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘〈𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3061 〈cop 4632 ↾ cres 5687 ‘cfv 6561 predc-bnj14 34702 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 df-iota 6514 df-fv 6569 df-bnj14 34703 |
| This theorem is referenced by: bnj1523 35085 |
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