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 30394. 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 1830 | . . . . . 6 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴 | |
4 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑥 𝐻 Fn 𝐴 | |
5 | nfra1 2925 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉) | |
6 | 3, 4, 5 | nf3an 1819 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘〈𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
7 | 2, 6 | nfxfr 1771 | . . . 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 30344 | . . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵) |
12 | 9, 11 | bnj1307 30345 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶) |
13 | 12 | nfcii 2742 | . . . . . . 7 ⊢ Ⅎ𝑥𝐶 |
14 | 13 | nfuni 4378 | . . . . . 6 ⊢ Ⅎ𝑥∪ 𝐶 |
15 | 8, 14 | nfcxfr 2749 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝐻 | |
17 | 15, 16 | nfne 2882 | . . . 4 ⊢ Ⅎ𝑥 𝐹 ≠ 𝐻 |
18 | 7, 17 | nfan 1816 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐹 ≠ 𝐻) |
19 | 1, 18 | nfxfr 1771 | . 2 ⊢ Ⅎ𝑥𝜓 |
20 | 19 | nf5ri 2053 | 1 ⊢ (𝜓 → ∀𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∀wal 1473 = wceq 1475 {cab 2596 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 〈cop 4131 ∪ cuni 4372 ↾ cres 5040 Fn wfn 5799 ‘cfv 5804 predc-bnj14 30007 FrSe w-bnj15 30011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-uni 4373 |
This theorem is referenced by: bnj1523 30393 |
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