Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnvintabd Structured version   Visualization version   GIF version

Theorem cnvintabd 39963
Description: Value of the converse of the intersection of a nonempty class. (Contributed by RP, 20-Aug-2020.)
Hypothesis
Ref Expression
cnvintabd.x (𝜑 → ∃𝑥𝜓)
Assertion
Ref Expression
cnvintabd (𝜑 {𝑥𝜓} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)})
Distinct variable groups:   𝜓,𝑤   𝑥,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑤)   𝜓(𝑥)

Proof of Theorem cnvintabd
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnvintabd.x . . . . . 6 (𝜑 → ∃𝑥𝜓)
2 pm5.5 364 . . . . . 6 (∃𝑥𝜓 → ((∃𝑥𝜓𝑦 ∈ (V × V)) ↔ 𝑦 ∈ (V × V)))
31, 2syl 17 . . . . 5 (𝜑 → ((∃𝑥𝜓𝑦 ∈ (V × V)) ↔ 𝑦 ∈ (V × V)))
43bicomd 225 . . . 4 (𝜑 → (𝑦 ∈ (V × V) ↔ (∃𝑥𝜓𝑦 ∈ (V × V))))
54anbi1d 631 . . 3 (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓𝑦𝑥)) ↔ ((∃𝑥𝜓𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓𝑦𝑥))))
6 elcnvintab 39962 . . 3 (𝑦 {𝑥𝜓} ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓𝑦𝑥)))
7 vex 3497 . . . . . 6 𝑥 ∈ V
87cnvex 7629 . . . . 5 𝑥 ∈ V
9 relcnv 5966 . . . . . 6 Rel 𝑥
10 df-rel 5561 . . . . . 6 (Rel 𝑥𝑥 ⊆ (V × V))
119, 10mpbi 232 . . . . 5 𝑥 ⊆ (V × V)
128, 11elmapintrab 39936 . . . 4 (𝑦 ∈ V → (𝑦 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)} ↔ ((∃𝑥𝜓𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓𝑦𝑥))))
1312elv 3499 . . 3 (𝑦 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)} ↔ ((∃𝑥𝜓𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓𝑦𝑥)))
145, 6, 133bitr4g 316 . 2 (𝜑 → (𝑦 {𝑥𝜓} ↔ 𝑦 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)}))
1514eqrdv 2819 1 (𝜑 {𝑥𝜓} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)})
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 3494  wss 3935  𝒫 cpw 4538   cint 4875   × cxp 5552  ccnv 5553  Rel wrel 5559
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 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
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 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-int 4876  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-1st 7688  df-2nd 7689
This theorem is referenced by:  clcnvlem  39983
  Copyright terms: Public domain W3C validator