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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj900 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32282. 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 |
---|---|
bnj900.3 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj900.4 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
Ref | Expression |
---|---|
bnj900 | ⊢ (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj900.4 | . . . . . 6 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
2 | 1 | bnj1436 32111 | . . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
3 | simp1 1132 | . . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → 𝑓 Fn 𝑛) | |
4 | 3 | reximi 3243 | . . . . 5 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∃𝑛 ∈ 𝐷 𝑓 Fn 𝑛) |
5 | fndm 6454 | . . . . . 6 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛) | |
6 | 5 | reximi 3243 | . . . . 5 ⊢ (∃𝑛 ∈ 𝐷 𝑓 Fn 𝑛 → ∃𝑛 ∈ 𝐷 dom 𝑓 = 𝑛) |
7 | 2, 4, 6 | 3syl 18 | . . . 4 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛 ∈ 𝐷 dom 𝑓 = 𝑛) |
8 | 7 | bnj1196 32066 | . . 3 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) |
9 | nfre1 3306 | . . . . . . 7 ⊢ Ⅎ𝑛∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) | |
10 | 9 | nfab 2984 | . . . . . 6 ⊢ Ⅎ𝑛{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
11 | 1, 10 | nfcxfr 2975 | . . . . 5 ⊢ Ⅎ𝑛𝐵 |
12 | 11 | nfcri 2971 | . . . 4 ⊢ Ⅎ𝑛 𝑓 ∈ 𝐵 |
13 | 12 | 19.37 2230 | . . 3 ⊢ (∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) ↔ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛))) |
14 | 8, 13 | mpbir 233 | . 2 ⊢ ∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) |
15 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑛∅ ∈ dom 𝑓 | |
16 | 12, 15 | nfim 1893 | . . 3 ⊢ Ⅎ𝑛(𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓) |
17 | bnj900.3 | . . . . . 6 ⊢ 𝐷 = (ω ∖ {∅}) | |
18 | 17 | bnj529 32012 | . . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛) |
19 | eleq2 2901 | . . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛)) | |
20 | 19 | biimparc 482 | . . . . 5 ⊢ ((∅ ∈ 𝑛 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓) |
21 | 18, 20 | sylan 582 | . . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓) |
22 | 21 | imim2i 16 | . . 3 ⊢ ((𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓)) |
23 | 16, 22 | exlimi 2213 | . 2 ⊢ (∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓)) |
24 | 14, 23 | ax-mp 5 | 1 ⊢ (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ∃wrex 3139 ∖ cdif 3932 ∅c0 4290 {csn 4566 dom cdm 5554 Fn wfn 6349 ωcom 7579 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-tr 5172 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-fn 6357 df-om 7580 |
This theorem is referenced by: bnj906 32202 |
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