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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1518 | 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 |
|---|---|
| bnj1518.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
| bnj1518.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
| bnj1518.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
| bnj1518.4 | ⊢ 𝐹 = ∪ 𝐶 |
| bnj1518.5 | ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)) |
| bnj1518.6 | ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓)) |
| Ref | Expression |
|---|---|
| bnj1518 | ⊢ (𝜓 → ∀𝑑𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1518.6 | . . 3 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓)) | |
| 2 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑑𝜑 | |
| 3 | bnj1518.3 | . . . . . 6 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
| 4 | nfre1 3285 | . . . . . . 7 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) | |
| 5 | 4 | nfab 2911 | . . . . . 6 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
| 6 | 3, 5 | nfcxfr 2903 | . . . . 5 ⊢ Ⅎ𝑑𝐶 |
| 7 | 6 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐶 |
| 8 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑑 𝑥 ∈ dom 𝑓 | |
| 9 | 2, 7, 8 | nf3an 1901 | . . 3 ⊢ Ⅎ𝑑(𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓) |
| 10 | 1, 9 | nfxfr 1853 | . 2 ⊢ Ⅎ𝑑𝜓 |
| 11 | 10 | nf5ri 2195 | 1 ⊢ (𝜓 → ∀𝑑𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 〈cop 4632 ∪ cuni 4907 dom cdm 5685 ↾ cres 5687 Fn wfn 6556 ‘cfv 6561 predc-bnj14 34702 FrSe w-bnj15 34706 |
| 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-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 |
| This theorem is referenced by: bnj1501 35081 |
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