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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfse3 | Structured version Visualization version GIF version |
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 5940 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) | |
2 | df-pred 6131 | . . . 4 ⊢ Pred(𝑅, 𝐴, 𝑥) = (𝐴 ∩ (◡𝑅 “ {𝑥})) | |
3 | 2 | eleq1i 2842 | . . 3 ⊢ (Pred(𝑅, 𝐴, 𝑥) ∈ V ↔ (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
4 | 3 | ralbii 3097 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
5 | 1, 4 | bitr4i 281 | 1 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2111 ∀wral 3070 Vcvv 3409 ∩ cin 3859 {csn 4525 Se wse 5485 ◡ccnv 5527 “ cima 5531 Predcpred 6130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-br 5037 df-opab 5099 df-se 5488 df-xp 5534 df-cnv 5536 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 |
This theorem is referenced by: sexp2 33360 sexp3 33366 |
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