Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj518 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj852 30245. 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 |
---|---|
bnj518.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
bnj518.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj518.3 | ⊢ (𝜏 ↔ (𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛)) |
Ref | Expression |
---|---|
bnj518 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj518.3 | . . . 4 ⊢ (𝜏 ↔ (𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛)) | |
2 | bnj334 30032 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛)) | |
3 | 1, 2 | bitri 263 | . . 3 ⊢ (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛)) |
4 | df-bnj17 30006 | . . . 4 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛) ↔ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑝 ∈ 𝑛)) | |
5 | bnj518.1 | . . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
6 | bnj518.2 | . . . . . 6 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
7 | 5, 6 | bnj517 30209 | . . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑝 ∈ 𝑛 (𝑓‘𝑝) ⊆ 𝐴) |
8 | 7 | r19.21bi 2916 | . . . 4 ⊢ (((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴) |
9 | 4, 8 | sylbi 206 | . . 3 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴) |
10 | 3, 9 | sylbi 206 | . 2 ⊢ (𝜏 → (𝑓‘𝑝) ⊆ 𝐴) |
11 | ssel 3562 | . . . 4 ⊢ ((𝑓‘𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓‘𝑝) → 𝑦 ∈ 𝐴)) | |
12 | bnj93 30187 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑦 ∈ 𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V) | |
13 | 12 | ex 449 | . . . 4 ⊢ (𝑅 FrSe 𝐴 → (𝑦 ∈ 𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V)) |
14 | 11, 13 | sylan9r 688 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓‘𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V)) |
15 | 14 | ralrimiv 2948 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
16 | 10, 15 | sylan2 490 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ∪ ciun 4455 suc csuc 5642 ‘cfv 5804 ωcom 6957 ∧ w-bnj17 30005 predc-bnj14 30007 FrSe w-bnj15 30011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fv 5812 df-om 6958 df-bnj17 30006 df-bnj14 30008 df-bnj13 30010 df-bnj15 30012 |
This theorem is referenced by: bnj535 30214 bnj546 30220 |
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