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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj553 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 31298. 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 |
---|---|
bnj553.1 | ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
bnj553.2 | ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj553.3 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj553.4 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)〉}) |
bnj553.5 | ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) |
bnj553.6 | ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚)) |
bnj553.7 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) |
bnj553.8 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, 𝐶〉}) |
bnj553.9 | ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) |
bnj553.10 | ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) |
bnj553.11 | ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) |
bnj553.12 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛) |
Ref | Expression |
---|---|
bnj553 | ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj553.12 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛) | |
2 | 1 | bnj930 31147 | . . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → Fun 𝐺) |
3 | opex 5081 | . . . . . . 7 ⊢ 〈𝑚, 𝐶〉 ∈ V | |
4 | 3 | snid 4353 | . . . . . 6 ⊢ 〈𝑚, 𝐶〉 ∈ {〈𝑚, 𝐶〉} |
5 | elun2 3924 | . . . . . 6 ⊢ (〈𝑚, 𝐶〉 ∈ {〈𝑚, 𝐶〉} → 〈𝑚, 𝐶〉 ∈ (𝑓 ∪ {〈𝑚, 𝐶〉})) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 〈𝑚, 𝐶〉 ∈ (𝑓 ∪ {〈𝑚, 𝐶〉}) |
7 | bnj553.8 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, 𝐶〉}) | |
8 | 6, 7 | eleqtrri 2838 | . . . 4 ⊢ 〈𝑚, 𝐶〉 ∈ 𝐺 |
9 | funopfv 6396 | . . . 4 ⊢ (Fun 𝐺 → (〈𝑚, 𝐶〉 ∈ 𝐺 → (𝐺‘𝑚) = 𝐶)) | |
10 | 2, 8, 9 | mpisyl 21 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝐺‘𝑚) = 𝐶) |
11 | 10 | 3ad2ant1 1128 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐶) |
12 | fveq2 6352 | . . . . . 6 ⊢ (𝑝 = 𝑖 → (𝐺‘𝑝) = (𝐺‘𝑖)) | |
13 | 12 | bnj1113 31163 | . . . . 5 ⊢ (𝑝 = 𝑖 → ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
14 | bnj553.11 | . . . . 5 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) | |
15 | bnj553.10 | . . . . 5 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) | |
16 | 13, 14, 15 | 3eqtr4g 2819 | . . . 4 ⊢ (𝑝 = 𝑖 → 𝐿 = 𝐾) |
17 | 16 | 3ad2ant3 1130 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐿 = 𝐾) |
18 | bnj553.5 | . . . . 5 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) | |
19 | bnj553.9 | . . . . 5 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) | |
20 | bnj553.4 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {〈𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)〉}) | |
21 | 18, 19, 15, 20, 1 | bnj548 31274 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝐵 = 𝐾) |
22 | 21 | 3adant3 1127 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐵 = 𝐾) |
23 | fveq2 6352 | . . . . . 6 ⊢ (𝑝 = 𝑖 → (𝑓‘𝑝) = (𝑓‘𝑖)) | |
24 | 23 | bnj1113 31163 | . . . . 5 ⊢ (𝑝 = 𝑖 → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
25 | bnj553.7 | . . . . . . 7 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) | |
26 | 19, 25 | eqeq12i 2774 | . . . . . 6 ⊢ (𝐵 = 𝐶 ↔ ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)) |
27 | eqcom 2767 | . . . . . 6 ⊢ (∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ↔ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) | |
28 | 26, 27 | bitri 264 | . . . . 5 ⊢ (𝐵 = 𝐶 ↔ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
29 | 24, 28 | sylibr 224 | . . . 4 ⊢ (𝑝 = 𝑖 → 𝐵 = 𝐶) |
30 | 29 | 3ad2ant3 1130 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐵 = 𝐶) |
31 | 17, 22, 30 | 3eqtr2rd 2801 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐶 = 𝐿) |
32 | 11, 31 | eqtrd 2794 | 1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ∖ cdif 3712 ∪ cun 3713 ∅c0 4058 {csn 4321 〈cop 4327 ∪ ciun 4672 suc csuc 5886 Fun wfun 6043 Fn wfn 6044 ‘cfv 6049 ωcom 7230 predc-bnj14 31063 FrSe w-bnj15 31067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-res 5278 df-iota 6012 df-fun 6051 df-fn 6052 df-fv 6057 |
This theorem is referenced by: bnj557 31278 |
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