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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elcnvcnvintab | Structured version Visualization version GIF version | ||
| Description: Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020.) |
| Ref | Expression |
|---|---|
| elcnvcnvintab | ⊢ (𝐴 ∈ ◡◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnvcnv 6212 | . . . 4 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) | |
| 2 | incom 4209 | . . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} ∩ (V × V)) = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) | |
| 3 | 1, 2 | eqtri 2765 | . . 3 ⊢ ◡◡∩ {𝑥 ∣ 𝜑} = ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) |
| 4 | 3 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ ◡◡∩ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ((V × V) ∩ ∩ {𝑥 ∣ 𝜑})) |
| 5 | elinintab 43588 | . 2 ⊢ (𝐴 ∈ ((V × V) ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) | |
| 6 | 4, 5 | bitri 275 | 1 ⊢ (𝐴 ∈ ◡◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∩ cin 3950 ∩ cint 4946 × cxp 5683 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-int 4947 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: cnvcnvintabd 43613 |
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