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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 31676. 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 31425 | . . 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 31627 | . . 3 ⊢ 𝐶 = 𝐾 |
8 | 2, 7 | syl6eleq 2916 | . 2 ⊢ (𝜑 → ℎ ∈ 𝐾) |
9 | 6 | abeq2i 2940 | . . . . . 6 ⊢ (ℎ ∈ 𝐾 ↔ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍))) |
10 | 9 | bnj1238 31423 | . . . . 5 ⊢ (ℎ ∈ 𝐾 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑) |
11 | 10 | bnj1196 31411 | . . . 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 | abeq2i 2940 | . . . . . 6 ⊢ (𝑑 ∈ 𝐵 ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)) |
14 | 13 | simplbi 493 | . . . . 5 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴) |
15 | fndm 6223 | . . . . 5 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑) | |
16 | 14, 15 | bnj1241 31424 | . . . 4 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → dom ℎ ⊆ 𝐴) |
17 | 11, 16 | bnj593 31361 | . . 3 ⊢ (ℎ ∈ 𝐾 → ∃𝑑dom ℎ ⊆ 𝐴) |
18 | 17 | bnj937 31388 | . 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 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 {cab 2811 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 {crab 3121 ∩ cin 3797 ⊆ wss 3798 〈cop 4403 class class class wbr 4873 dom cdm 5342 ↾ cres 5344 Fn wfn 6118 ‘cfv 6123 ∧ w-bnj17 31301 predc-bnj14 31303 FrSe w-bnj15 31307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-res 5354 df-iota 6086 df-fun 6125 df-fn 6126 df-fv 6131 df-bnj17 31302 |
This theorem is referenced by: (None) |
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