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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj229 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj517 32152. 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 |
---|---|
bnj229.1 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
Ref | Expression |
---|---|
bnj229 | ⊢ ((𝑛 ∈ 𝑁 ∧ (suc 𝑚 = 𝑛 ∧ 𝑚 ∈ ω ∧ 𝜓)) → (𝐹‘𝑛) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj213 32149 | . . 3 ⊢ pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 | |
2 | 1 | bnj226 31999 | . 2 ⊢ ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 |
3 | bnj229.1 | . . . . . . . 8 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
4 | 3 | bnj222 32150 | . . . . . . 7 ⊢ (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))) |
5 | 4 | bnj228 32000 | . . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝜓) → (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))) |
6 | 5 | adantl 484 | . . . . 5 ⊢ ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))) |
7 | eleq1 2900 | . . . . . . 7 ⊢ (suc 𝑚 = 𝑛 → (suc 𝑚 ∈ 𝑁 ↔ 𝑛 ∈ 𝑁)) | |
8 | fveqeq2 6673 | . . . . . . 7 ⊢ (suc 𝑚 = 𝑛 → ((𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))) | |
9 | 7, 8 | imbi12d 347 | . . . . . 6 ⊢ (suc 𝑚 = 𝑛 → ((suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛 ∈ 𝑁 → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))) |
10 | 9 | adantr 483 | . . . . 5 ⊢ ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → ((suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛 ∈ 𝑁 → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))) |
11 | 6, 10 | mpbid 234 | . . . 4 ⊢ ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (𝑛 ∈ 𝑁 → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))) |
12 | 11 | 3impb 1111 | . . 3 ⊢ ((suc 𝑚 = 𝑛 ∧ 𝑚 ∈ ω ∧ 𝜓) → (𝑛 ∈ 𝑁 → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))) |
13 | 12 | impcom 410 | . 2 ⊢ ((𝑛 ∈ 𝑁 ∧ (suc 𝑚 = 𝑛 ∧ 𝑚 ∈ ω ∧ 𝜓)) → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)) |
14 | 2, 13 | bnj1262 32077 | 1 ⊢ ((𝑛 ∈ 𝑁 ∧ (suc 𝑚 = 𝑛 ∧ 𝑚 ∈ ω ∧ 𝜓)) → (𝐹‘𝑛) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ∪ ciun 4911 suc csuc 6187 ‘cfv 6349 ωcom 7574 predc-bnj14 31953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-suc 6191 df-iota 6308 df-fv 6357 df-bnj14 31954 |
This theorem is referenced by: (None) |
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