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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj900 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 31691. 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 31523 | . . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
3 | simp1 1127 | . . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → 𝑓 Fn 𝑛) | |
4 | 3 | reximi 3191 | . . . . 5 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∃𝑛 ∈ 𝐷 𝑓 Fn 𝑛) |
5 | fndm 6235 | . . . . . 6 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛) | |
6 | 5 | reximi 3191 | . . . . 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 31478 | . . 3 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) |
9 | nfre1 3185 | . . . . . . 7 ⊢ Ⅎ𝑛∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) | |
10 | 9 | nfab 2939 | . . . . . 6 ⊢ Ⅎ𝑛{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
11 | 1, 10 | nfcxfr 2931 | . . . . 5 ⊢ Ⅎ𝑛𝐵 |
12 | 11 | nfcri 2928 | . . . 4 ⊢ Ⅎ𝑛 𝑓 ∈ 𝐵 |
13 | 12 | 19.37 2217 | . . 3 ⊢ (∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) ↔ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛))) |
14 | 8, 13 | mpbir 223 | . 2 ⊢ ∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) |
15 | nfv 1957 | . . . 4 ⊢ Ⅎ𝑛∅ ∈ dom 𝑓 | |
16 | 12, 15 | nfim 1943 | . . 3 ⊢ Ⅎ𝑛(𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓) |
17 | bnj900.3 | . . . . . 6 ⊢ 𝐷 = (ω ∖ {∅}) | |
18 | 17 | bnj529 31424 | . . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛) |
19 | eleq2 2847 | . . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛)) | |
20 | 19 | biimparc 473 | . . . . 5 ⊢ ((∅ ∈ 𝑛 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓) |
21 | 18, 20 | sylan 575 | . . . 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 2202 | . 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 386 ∧ w3a 1071 = wceq 1601 ∃wex 1823 ∈ wcel 2106 {cab 2762 ∃wrex 3090 ∖ cdif 3788 ∅c0 4140 {csn 4397 dom cdm 5355 Fn wfn 6130 ωcom 7343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-tr 4988 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-fn 6138 df-om 7344 |
This theorem is referenced by: bnj906 31613 |
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