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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1307 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 31130. 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 2755 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
4 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑑 | |
5 | nfra1 2941 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) | |
6 | 4, 5 | nfan 1828 | . . . . 5 ⊢ Ⅎ𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) |
7 | 3, 6 | nfrex 3007 | . . . 4 ⊢ Ⅎ𝑥∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) |
8 | 7 | nfab 2769 | . . 3 ⊢ Ⅎ𝑥{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
9 | 1, 8 | nfcxfr 2762 | . 2 ⊢ Ⅎ𝑥𝐶 |
10 | 9 | nfcrii 2757 | 1 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 ∃wrex 2913 Fn wfn 5883 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 |
This theorem is referenced by: bnj1311 31092 bnj1373 31098 bnj1498 31129 bnj1525 31137 bnj1523 31139 |
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