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 31140. 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 1843 | . . . . . 6 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴 | |
4 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑥 𝐻 Fn 𝐴 | |
5 | nfra1 2941 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉) | |
6 | 3, 4, 5 | nf3an 1831 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
7 | 2, 6 | nfxfr 1779 | . . . 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 31090 | . . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵) |
12 | 9, 11 | bnj1307 31091 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶) |
13 | 12 | nfcii 2755 | . . . . . . 7 ⊢ Ⅎ𝑥𝐶 |
14 | 13 | nfuni 4442 | . . . . . 6 ⊢ Ⅎ𝑥∪ 𝐶 |
15 | 8, 14 | nfcxfr 2762 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑥𝐻 | |
17 | 15, 16 | nfne 2894 | . . . 4 ⊢ Ⅎ𝑥 𝐹 ≠ 𝐻 |
18 | 7, 17 | nfan 1828 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐹 ≠ 𝐻) |
19 | 1, 18 | nfxfr 1779 | . 2 ⊢ Ⅎ𝑥𝜓 |
20 | 19 | nf5ri 2065 | 1 ⊢ (𝜓 → ∀𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 {cab 2608 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 〈cop 4183 ∪ cuni 4436 ↾ cres 5116 Fn wfn 5883 ‘cfv 5888 predc-bnj14 30754 FrSe w-bnj15 30758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-uni 4437 |
This theorem is referenced by: bnj1523 31139 |
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