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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1245 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32709. 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 32455 | . . 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 32660 | . . 3 ⊢ 𝐶 = 𝐾 |
8 | 2, 7 | eleqtrdi 2841 | . 2 ⊢ (𝜑 → ℎ ∈ 𝐾) |
9 | 6 | abeq2i 2865 | . . . . . 6 ⊢ (ℎ ∈ 𝐾 ↔ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍))) |
10 | 9 | bnj1238 32453 | . . . . 5 ⊢ (ℎ ∈ 𝐾 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑) |
11 | 10 | bnj1196 32441 | . . . 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 2865 | . . . . . 6 ⊢ (𝑑 ∈ 𝐵 ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)) |
14 | 13 | simplbi 501 | . . . . 5 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴) |
15 | fndm 6459 | . . . . 5 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑) | |
16 | 14, 15 | bnj1241 32454 | . . . 4 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → dom ℎ ⊆ 𝐴) |
17 | 11, 16 | bnj593 32391 | . . 3 ⊢ (ℎ ∈ 𝐾 → ∃𝑑dom ℎ ⊆ 𝐴) |
18 | 17 | bnj937 32418 | . 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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 {cab 2714 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 {crab 3055 ∩ cin 3852 ⊆ wss 3853 〈cop 4533 class class class wbr 5039 dom cdm 5536 ↾ cres 5538 Fn wfn 6353 ‘cfv 6358 ∧ w-bnj17 32331 predc-bnj14 32333 FrSe w-bnj15 32337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-res 5548 df-iota 6316 df-fun 6360 df-fn 6361 df-fv 6366 df-bnj17 32332 |
This theorem is referenced by: (None) |
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