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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1519 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj1500 35082. 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 |
|---|---|
| bnj1519.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
| bnj1519.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
| bnj1519.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
| bnj1519.4 | ⊢ 𝐹 = ∪ 𝐶 |
| Ref | Expression |
|---|---|
| bnj1519 | ⊢ ((𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) → ∀𝑑(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1519.4 | . . . . 5 ⊢ 𝐹 = ∪ 𝐶 | |
| 2 | bnj1519.3 | . . . . . . 7 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
| 3 | nfre1 3285 | . . . . . . . 8 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) | |
| 4 | 3 | nfab 2911 | . . . . . . 7 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
| 5 | 2, 4 | nfcxfr 2903 | . . . . . 6 ⊢ Ⅎ𝑑𝐶 |
| 6 | 5 | nfuni 4914 | . . . . 5 ⊢ Ⅎ𝑑∪ 𝐶 |
| 7 | 1, 6 | nfcxfr 2903 | . . . 4 ⊢ Ⅎ𝑑𝐹 |
| 8 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑑𝑥 | |
| 9 | 7, 8 | nffv 6916 | . . 3 ⊢ Ⅎ𝑑(𝐹‘𝑥) |
| 10 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑑𝐺 | |
| 11 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑑 pred(𝑥, 𝐴, 𝑅) | |
| 12 | 7, 11 | nfres 5999 | . . . . 5 ⊢ Ⅎ𝑑(𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) |
| 13 | 8, 12 | nfop 4889 | . . . 4 ⊢ Ⅎ𝑑〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
| 14 | 10, 13 | nffv 6916 | . . 3 ⊢ Ⅎ𝑑(𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
| 15 | 9, 14 | nfeq 2919 | . 2 ⊢ Ⅎ𝑑(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
| 16 | 15 | nf5ri 2195 | 1 ⊢ ((𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) → ∀𝑑(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 {cab 2714 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 〈cop 4632 ∪ cuni 4907 ↾ cres 5687 Fn wfn 6556 ‘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 |
| This theorem is referenced by: bnj1501 35081 |
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