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 32346. 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 1915 | . . . . . 6 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴 | |
4 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑥 𝐻 Fn 𝐴 | |
5 | nfra1 3221 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉) | |
6 | 3, 4, 5 | nf3an 1902 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
7 | 2, 6 | nfxfr 1853 | . . . 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 32296 | . . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵) |
12 | 9, 11 | bnj1307 32297 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶) |
13 | 12 | nfcii 2967 | . . . . . . 7 ⊢ Ⅎ𝑥𝐶 |
14 | 13 | nfuni 4847 | . . . . . 6 ⊢ Ⅎ𝑥∪ 𝐶 |
15 | 8, 14 | nfcxfr 2977 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑥𝐻 | |
17 | 15, 16 | nfne 3121 | . . . 4 ⊢ Ⅎ𝑥 𝐹 ≠ 𝐻 |
18 | 7, 17 | nfan 1900 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐹 ≠ 𝐻) |
19 | 1, 18 | nfxfr 1853 | . 2 ⊢ Ⅎ𝑥𝜓 |
20 | 19 | nf5ri 2195 | 1 ⊢ (𝜓 → ∀𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∀wal 1535 = wceq 1537 {cab 2801 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 〈cop 4575 ∪ cuni 4840 ↾ cres 5559 Fn wfn 6352 ‘cfv 6357 predc-bnj14 31960 FrSe w-bnj15 31964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-uni 4841 |
This theorem is referenced by: bnj1523 32345 |
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