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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 | pm5.5 364 | . . . . . 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 225 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ (V × V) ↔ (∃𝑥𝜓 → 𝑦 ∈ (V × V)))) |
5 | 4 | anbi1d 631 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥)) ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥)))) |
6 | elcnvintab 39955 | . . 3 ⊢ (𝑦 ∈ ◡∩ {𝑥 ∣ 𝜓} ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥))) | |
7 | vex 3498 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 7 | cnvex 7624 | . . . . 5 ⊢ ◡𝑥 ∈ V |
9 | relcnv 5962 | . . . . . 6 ⊢ Rel ◡𝑥 | |
10 | df-rel 5557 | . . . . . 6 ⊢ (Rel ◡𝑥 ↔ ◡𝑥 ⊆ (V × V)) | |
11 | 9, 10 | mpbi 232 | . . . . 5 ⊢ ◡𝑥 ⊆ (V × V) |
12 | 8, 11 | elmapintrab 39929 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥)))) |
13 | 12 | elv 3500 | . . 3 ⊢ (𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)} ↔ ((∃𝑥𝜓 → 𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓 → 𝑦 ∈ ◡𝑥))) |
14 | 5, 6, 13 | 3bitr4g 316 | . 2 ⊢ (𝜑 → (𝑦 ∈ ◡∩ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)})) |
15 | 14 | eqrdv 2819 | 1 ⊢ (𝜑 → ◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 {crab 3142 Vcvv 3495 ⊆ wss 3936 𝒫 cpw 4539 ∩ cint 4869 × cxp 5548 ◡ccnv 5549 Rel wrel 5555 |
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-pow 5259 ax-pr 5322 ax-un 7455 |
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-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-int 4870 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fv 6358 df-1st 7683 df-2nd 7684 |
This theorem is referenced by: clcnvlem 39976 |
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