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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-seex | Structured version Visualization version GIF version | ||
| Description: Version of seex 5642 with a disjoint variable condition replaced by a nonfreeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.) |
| Ref | Expression |
|---|---|
| bj-seex.nf | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| bj-seex | ⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-se 5636 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V) | |
| 2 | bj-seex.nf | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
| 3 | 2 | nfeq2 2922 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
| 4 | breq2 5145 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵)) | |
| 5 | 3, 4 | rabbid 3463 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵}) |
| 6 | 5 | eleq1d 2825 | . . 3 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V)) |
| 7 | 6 | rspccva 3620 | . 2 ⊢ ((∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V) |
| 8 | 1, 7 | sylanb 581 | 1 ⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2889 ∀wral 3060 {crab 3435 Vcvv 3479 class class class wbr 5141 Se wse 5633 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5142 df-se 5636 |
| This theorem is referenced by: (None) |
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