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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 34726. 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 2883 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
4 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑑 | |
5 | nfra1 3279 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) | |
6 | 4, 5 | nfan 1894 | . . . . 5 ⊢ Ⅎ𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) |
7 | 3, 6 | nfrexw 3308 | . . . 4 ⊢ Ⅎ𝑥∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) |
8 | 7 | nfab 2905 | . . 3 ⊢ Ⅎ𝑥{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
9 | 1, 8 | nfcxfr 2897 | . 2 ⊢ Ⅎ𝑥𝐶 |
10 | 9 | nfcrii 2891 | 1 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1531 = wceq 1533 ∈ wcel 2098 {cab 2705 ∀wral 3058 ∃wrex 3067 Fn wfn 6548 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 |
This theorem is referenced by: bnj1311 34688 bnj1373 34694 bnj1498 34725 bnj1525 34733 bnj1523 34735 |
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