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| Description: Alternate definition of set-like relationships. (Contributed by Scott Fenton, 19-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| dfse3 | ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfse2 6117 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) | |
| 2 | df-pred 6320 | . . . 4 ⊢ Pred(𝑅, 𝐴, 𝑥) = (𝐴 ∩ (◡𝑅 “ {𝑥})) | |
| 3 | 2 | eleq1i 2831 | . . 3 ⊢ (Pred(𝑅, 𝐴, 𝑥) ∈ V ↔ (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) | 
| 4 | 3 | ralbii 3092 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) | 
| 5 | 1, 4 | bitr4i 278 | 1 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ∩ cin 3949 {csn 4625 Se wse 5634 ◡ccnv 5683 “ cima 5687 Predcpred 6319 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-se 5637 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 | 
| This theorem is referenced by: sexp2 8172 sexp3 8179 ttrclselem2 9767 ttrclse 9768 | 
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