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Mirrors > Home > NFE Home > Th. List > elimakg | GIF version |
Description: Membership in a Kuratowski image. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
elimakg | ⊢ (C ∈ V → (C ∈ (A “k B) ↔ ∃y ∈ B ⟪y, C⟫ ∈ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opkeq2 4060 | . . . 4 ⊢ (x = C → ⟪y, x⟫ = ⟪y, C⟫) | |
2 | 1 | eleq1d 2419 | . . 3 ⊢ (x = C → (⟪y, x⟫ ∈ A ↔ ⟪y, C⟫ ∈ A)) |
3 | 2 | rexbidv 2635 | . 2 ⊢ (x = C → (∃y ∈ B ⟪y, x⟫ ∈ A ↔ ∃y ∈ B ⟪y, C⟫ ∈ A)) |
4 | df-imak 4189 | . 2 ⊢ (A “k B) = {x ∣ ∃y ∈ B ⟪y, x⟫ ∈ A} | |
5 | 3, 4 | elab2g 2987 | 1 ⊢ (C ∈ V → (C ∈ (A “k B) ↔ ∃y ∈ B ⟪y, C⟫ ∈ A)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ⟪copk 4057 “k cimak 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 df-opk 4058 df-imak 4189 |
This theorem is referenced by: elimakvg 4258 elimak 4259 |
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