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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1245 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 34726. 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 |
---|---|
bnj1245.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1245.2 | ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ |
bnj1245.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1245.4 | ⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ) |
bnj1245.5 | ⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)} |
bnj1245.6 | ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))) |
bnj1245.7 | ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥)) |
bnj1245.8 | ⊢ 𝑍 = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩ |
bnj1245.9 | ⊢ 𝐾 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍))} |
Ref | Expression |
---|---|
bnj1245 | ⊢ (𝜑 → dom ℎ ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1245.6 | . . . 4 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷))) | |
2 | 1 | bnj1247 34472 | . . 3 ⊢ (𝜑 → ℎ ∈ 𝐶) |
3 | bnj1245.2 | . . . 4 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ | |
4 | bnj1245.3 | . . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
5 | bnj1245.8 | . . . 4 ⊢ 𝑍 = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩ | |
6 | bnj1245.9 | . . . 4 ⊢ 𝐾 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍))} | |
7 | 3, 4, 5, 6 | bnj1234 34677 | . . 3 ⊢ 𝐶 = 𝐾 |
8 | 2, 7 | eleqtrdi 2839 | . 2 ⊢ (𝜑 → ℎ ∈ 𝐾) |
9 | 6 | eqabri 2873 | . . . . . 6 ⊢ (ℎ ∈ 𝐾 ↔ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍))) |
10 | 9 | bnj1238 34470 | . . . . 5 ⊢ (ℎ ∈ 𝐾 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑) |
11 | 10 | bnj1196 34458 | . . . 4 ⊢ (ℎ ∈ 𝐾 → ∃𝑑(𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑)) |
12 | bnj1245.1 | . . . . . . 7 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | |
13 | 12 | eqabri 2873 | . . . . . 6 ⊢ (𝑑 ∈ 𝐵 ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)) |
14 | 13 | simplbi 496 | . . . . 5 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴) |
15 | fndm 6662 | . . . . 5 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑) | |
16 | 14, 15 | bnj1241 34471 | . . . 4 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → dom ℎ ⊆ 𝐴) |
17 | 11, 16 | bnj593 34409 | . . 3 ⊢ (ℎ ∈ 𝐾 → ∃𝑑dom ℎ ⊆ 𝐴) |
18 | 17 | bnj937 34435 | . 2 ⊢ (ℎ ∈ 𝐾 → dom ℎ ⊆ 𝐴) |
19 | 8, 18 | syl 17 | 1 ⊢ (𝜑 → dom ℎ ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {cab 2705 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3430 ∩ cin 3948 ⊆ wss 3949 ⟨cop 4638 class class class wbr 5152 dom cdm 5682 ↾ cres 5684 Fn wfn 6548 ‘cfv 6553 ∧ w-bnj17 34350 predc-bnj14 34352 FrSe w-bnj15 34356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-res 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 df-bnj17 34351 |
This theorem is referenced by: (None) |
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