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Mirrors > Home > MPE Home > Th. List > dfse2 | Structured version Visualization version GIF version |
Description: Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.) |
Ref | Expression |
---|---|
dfse2 | ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-se 5545 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
2 | dfrab3 4243 | . . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥}) | |
3 | iniseg 6005 | . . . . . . 7 ⊢ (𝑥 ∈ V → (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥}) | |
4 | 3 | elv 3438 | . . . . . 6 ⊢ (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥} |
5 | 4 | ineq2i 4143 | . . . . 5 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥}) |
6 | 2, 5 | eqtr4i 2769 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ (◡𝑅 “ {𝑥})) |
7 | 6 | eleq1i 2829 | . . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
8 | 7 | ralbii 3092 | . 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
9 | 1, 8 | bitri 274 | 1 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 ∈ wcel 2106 {cab 2715 ∀wral 3064 {crab 3068 Vcvv 3432 ∩ cin 3886 {csn 4561 class class class wbr 5074 Se wse 5542 ◡ccnv 5588 “ cima 5592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-se 5545 df-xp 5595 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 |
This theorem is referenced by: dfse3 6239 isoselem 7212 fnse 7974 |
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