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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-seex | Structured version Visualization version GIF version |
Description: Version of seex 5520 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 5517 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V) | |
2 | bj-seex.nf | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
3 | 2 | nfeq2 2997 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
4 | breq2 5072 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵)) | |
5 | 3, 4 | rabbid 3477 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} = {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵}) |
6 | 5 | eleq1d 2899 | . . 3 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V)) |
7 | 6 | rspccva 3624 | . 2 ⊢ ((∀𝑦 ∈ 𝐴 {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝑦} ∈ V ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V) |
8 | 1, 7 | sylanb 583 | 1 ⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 ∀wral 3140 {crab 3144 Vcvv 3496 class class class wbr 5068 Se wse 5514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-se 5517 |
This theorem is referenced by: (None) |
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