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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1518 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1500 34110. 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 1918 | . . . 4 ⊢ Ⅎ𝑑𝜑 | |
3 | bnj1518.3 | . . . . . 6 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
4 | nfre1 3283 | . . . . . . 7 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) | |
5 | 4 | nfab 2910 | . . . . . 6 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
6 | 3, 5 | nfcxfr 2902 | . . . . 5 ⊢ Ⅎ𝑑𝐶 |
7 | 6 | nfcri 2891 | . . . 4 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐶 |
8 | nfv 1918 | . . . 4 ⊢ Ⅎ𝑑 𝑥 ∈ dom 𝑓 | |
9 | 2, 7, 8 | nf3an 1905 | . . 3 ⊢ Ⅎ𝑑(𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓) |
10 | 1, 9 | nfxfr 1856 | . 2 ⊢ Ⅎ𝑑𝜓 |
11 | 10 | nf5ri 2189 | 1 ⊢ (𝜓 → ∀𝑑𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 ⟨cop 4635 ∪ cuni 4909 dom cdm 5677 ↾ cres 5679 Fn wfn 6539 ‘cfv 6544 predc-bnj14 33730 FrSe w-bnj15 33734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rex 3072 |
This theorem is referenced by: bnj1501 34109 |
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