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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj554 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 31326. 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 |
---|---|
bnj554.19 | ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)) |
bnj554.20 | ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖)) |
bnj554.21 | ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) |
bnj554.22 | ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) |
bnj554.23 | ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) |
bnj554.24 | ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) |
Ref | Expression |
---|---|
bnj554 | ⊢ ((𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj554.19 | . . 3 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)) | |
2 | 1 | bnj1254 31215 | . 2 ⊢ (𝜂 → 𝑚 = suc 𝑝) |
3 | bnj554.20 | . . 3 ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖)) | |
4 | 3 | simp3bi 1141 | . 2 ⊢ (𝜁 → 𝑚 = suc 𝑖) |
5 | simpr 471 | . . 3 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑚 = suc 𝑖) | |
6 | bnj551 31147 | . . 3 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖) | |
7 | fveq2 6332 | . . . 4 ⊢ (𝑚 = suc 𝑖 → (𝐺‘𝑚) = (𝐺‘suc 𝑖)) | |
8 | fveq2 6332 | . . . . 5 ⊢ (𝑝 = 𝑖 → (𝐺‘𝑝) = (𝐺‘𝑖)) | |
9 | iuneq1 4668 | . . . . . 6 ⊢ ((𝐺‘𝑝) = (𝐺‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) | |
10 | bnj554.24 | . . . . . 6 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) | |
11 | bnj554.23 | . . . . . 6 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) | |
12 | 9, 10, 11 | 3eqtr4g 2830 | . . . . 5 ⊢ ((𝐺‘𝑝) = (𝐺‘𝑖) → 𝐿 = 𝐾) |
13 | 8, 12 | syl 17 | . . . 4 ⊢ (𝑝 = 𝑖 → 𝐿 = 𝐾) |
14 | 7, 13 | eqeqan12d 2787 | . . 3 ⊢ ((𝑚 = suc 𝑖 ∧ 𝑝 = 𝑖) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾)) |
15 | 5, 6, 14 | syl2anc 565 | . 2 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾)) |
16 | 2, 4, 15 | syl2an 575 | 1 ⊢ ((𝜂 ∧ 𝜁) → ((𝐺‘𝑚) = 𝐿 ↔ (𝐺‘suc 𝑖) = 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∪ ciun 4654 suc csuc 5868 ‘cfv 6031 ωcom 7211 ∧ w-bnj17 31089 predc-bnj14 31091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-un 7095 ax-reg 8652 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-eprel 5162 df-fr 5208 df-suc 5872 df-iota 5994 df-fv 6039 df-bnj17 31090 |
This theorem is referenced by: bnj558 31307 |
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