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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 5510 | . 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | |
2 | dfrab3 4278 | . . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥}) | |
3 | iniseg 5955 | . . . . . . 7 ⊢ (𝑥 ∈ V → (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥}) | |
4 | 3 | elv 3500 | . . . . . 6 ⊢ (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥} |
5 | 4 | ineq2i 4186 | . . . . 5 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥}) |
6 | 2, 5 | eqtr4i 2847 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ (◡𝑅 “ {𝑥})) |
7 | 6 | eleq1i 2903 | . . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
8 | 7 | ralbii 3165 | . 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
9 | 1, 8 | bitri 277 | 1 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 {crab 3142 Vcvv 3495 ∩ cin 3935 {csn 4561 class class class wbr 5059 Se wse 5507 ◡ccnv 5549 “ cima 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-se 5510 df-xp 5556 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 |
This theorem is referenced by: isoselem 7088 fnse 7821 |
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