Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1525 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1522 32241. 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 |
---|---|
bnj1525.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1525.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1525.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1525.4 | ⊢ 𝐹 = ∪ 𝐶 |
bnj1525.5 | ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉))) |
bnj1525.6 | ⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻)) |
Ref | Expression |
---|---|
bnj1525 | ⊢ (𝜓 → ∀𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1525.6 | . . 3 ⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻)) | |
2 | bnj1525.5 | . . . . 5 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉))) | |
3 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴 | |
4 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑥 𝐻 Fn 𝐴 | |
5 | nfra1 3216 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉) | |
6 | 3, 4, 5 | nf3an 1893 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
7 | 2, 6 | nfxfr 1844 | . . . 4 ⊢ Ⅎ𝑥𝜑 |
8 | bnj1525.4 | . . . . . 6 ⊢ 𝐹 = ∪ 𝐶 | |
9 | bnj1525.3 | . . . . . . . . 9 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
10 | bnj1525.1 | . . . . . . . . . 10 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | |
11 | 10 | bnj1309 32191 | . . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵) |
12 | 9, 11 | bnj1307 32192 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶) |
13 | 12 | nfcii 2962 | . . . . . . 7 ⊢ Ⅎ𝑥𝐶 |
14 | 13 | nfuni 4837 | . . . . . 6 ⊢ Ⅎ𝑥∪ 𝐶 |
15 | 8, 14 | nfcxfr 2972 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝐻 | |
17 | 15, 16 | nfne 3116 | . . . 4 ⊢ Ⅎ𝑥 𝐹 ≠ 𝐻 |
18 | 7, 17 | nfan 1891 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐹 ≠ 𝐻) |
19 | 1, 18 | nfxfr 1844 | . 2 ⊢ Ⅎ𝑥𝜓 |
20 | 19 | nf5ri 2185 | 1 ⊢ (𝜓 → ∀𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 ∀wal 1526 = wceq 1528 {cab 2796 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 〈cop 4563 ∪ cuni 4830 ↾ cres 5550 Fn wfn 6343 ‘cfv 6348 predc-bnj14 31857 FrSe w-bnj15 31861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-uni 4831 |
This theorem is referenced by: bnj1523 32240 |
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