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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1307 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 33731. 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 |
---|---|
bnj1307.1 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1307.2 | ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵) |
Ref | Expression |
---|---|
bnj1307 | ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1307.1 | . . 3 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
2 | bnj1307.2 | . . . . . 6 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵) | |
3 | 2 | nfcii 2888 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
4 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑑 | |
5 | nfra1 3266 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) | |
6 | 4, 5 | nfan 1903 | . . . . 5 ⊢ Ⅎ𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) |
7 | 3, 6 | nfrexw 3295 | . . . 4 ⊢ Ⅎ𝑥∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) |
8 | 7 | nfab 2910 | . . 3 ⊢ Ⅎ𝑥{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
9 | 1, 8 | nfcxfr 2902 | . 2 ⊢ Ⅎ𝑥𝐶 |
10 | 9 | nfcrii 2896 | 1 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3061 ∃wrex 3070 Fn wfn 6492 ‘cfv 6497 |
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-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 |
This theorem is referenced by: bnj1311 33693 bnj1373 33699 bnj1498 33730 bnj1525 33738 bnj1523 33740 |
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