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 32761. 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 1922 | . . . 4 ⊢ Ⅎ𝑑𝜑 | |
3 | bnj1518.3 | . . . . . 6 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
4 | nfre1 3225 | . . . . . . 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 1922 | . . . 4 ⊢ Ⅎ𝑑 𝑥 ∈ dom 𝑓 | |
9 | 2, 7, 8 | nf3an 1909 | . . 3 ⊢ Ⅎ𝑑(𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓) |
10 | 1, 9 | nfxfr 1860 | . 2 ⊢ Ⅎ𝑑𝜓 |
11 | 10 | nf5ri 2193 | 1 ⊢ (𝜓 → ∀𝑑𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∀wal 1541 = wceq 1543 ∈ wcel 2110 {cab 2714 ∀wral 3061 ∃wrex 3062 ⊆ wss 3866 〈cop 4547 ∪ cuni 4819 dom cdm 5551 ↾ cres 5553 Fn wfn 6375 ‘cfv 6380 predc-bnj14 32379 FrSe w-bnj15 32383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-rex 3067 |
This theorem is referenced by: bnj1501 32760 |
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