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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj900 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32392. 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 32221 | . . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
3 | simp1 1133 | . . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → 𝑓 Fn 𝑛) | |
4 | 3 | reximi 3206 | . . . . 5 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∃𝑛 ∈ 𝐷 𝑓 Fn 𝑛) |
5 | fndm 6425 | . . . . . 6 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛) | |
6 | 5 | reximi 3206 | . . . . 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 32176 | . . 3 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) |
9 | nfre1 3265 | . . . . . . 7 ⊢ Ⅎ𝑛∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) | |
10 | 9 | nfab 2961 | . . . . . 6 ⊢ Ⅎ𝑛{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
11 | 1, 10 | nfcxfr 2953 | . . . . 5 ⊢ Ⅎ𝑛𝐵 |
12 | 11 | nfcri 2943 | . . . 4 ⊢ Ⅎ𝑛 𝑓 ∈ 𝐵 |
13 | 12 | 19.37 2232 | . . 3 ⊢ (∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) ↔ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛))) |
14 | 8, 13 | mpbir 234 | . 2 ⊢ ∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) |
15 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑛∅ ∈ dom 𝑓 | |
16 | 12, 15 | nfim 1897 | . . 3 ⊢ Ⅎ𝑛(𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓) |
17 | bnj900.3 | . . . . . 6 ⊢ 𝐷 = (ω ∖ {∅}) | |
18 | 17 | bnj529 32122 | . . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛) |
19 | eleq2 2878 | . . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛)) | |
20 | 19 | biimparc 483 | . . . . 5 ⊢ ((∅ ∈ 𝑛 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓) |
21 | 18, 20 | sylan 583 | . . . 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 2215 | . 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 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 ∃wrex 3107 ∖ cdif 3878 ∅c0 4243 {csn 4525 dom cdm 5519 Fn wfn 6319 ωcom 7560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-fn 6327 df-om 7561 |
This theorem is referenced by: bnj906 32312 |
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