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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvintabd | Structured version Visualization version GIF version |
Description: Value of the converse of the intersection of a nonempty class. (Contributed by RP, 20-Aug-2020.) |
Ref | Expression |
---|---|
cnvintabd.x | ⊢ (𝜑 → ∃𝑥𝜓) |
Ref | Expression |
---|---|
cnvintabd | ⊢ (𝜑 → ◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvintabd.x | . . . . . 6 ⊢ (𝜑 → ∃𝑥𝜓) | |
2 | pm5.5 363 | . . . . . 6 ⊢ (∃𝑥𝜓 → ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ↔ 𝑦 ∈ (V × V))) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ↔ 𝑦 ∈ (V × V))) |
4 | 3 | bicomd 224 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ (V × V) ↔ (∃𝑥𝜓 → 𝑦 ∈ (V × V)))) |
5 | 4 | anbi1d 629 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥)) ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥)))) |
6 | elcnvintab 39840 | . . 3 ⊢ (𝑦 ∈ ◡∩ {𝑥 ∣ 𝜓} ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥))) | |
7 | vex 3495 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 7 | cnvex 7619 | . . . . 5 ⊢ ◡𝑥 ∈ V |
9 | relcnv 5960 | . . . . . 6 ⊢ Rel ◡𝑥 | |
10 | df-rel 5555 | . . . . . 6 ⊢ (Rel ◡𝑥 ↔ ◡𝑥 ⊆ (V × V)) | |
11 | 9, 10 | mpbi 231 | . . . . 5 ⊢ ◡𝑥 ⊆ (V × V) |
12 | 8, 11 | elmapintrab 39814 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥)))) |
13 | 12 | elv 3497 | . . 3 ⊢ (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥))) |
14 | 5, 6, 13 | 3bitr4g 315 | . 2 ⊢ (𝜑 → (𝑦 ∈ ◡∩ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)})) |
15 | 14 | eqrdv 2816 | 1 ⊢ (𝜑 → ◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 ∈ wcel 2105 {cab 2796 {crab 3139 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 ∩ cint 4867 × cxp 5546 ◡ccnv 5547 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7678 df-2nd 7679 |
This theorem is referenced by: clcnvlem 39861 |
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