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Theorem cnvintabd 38403
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 pm5.5 352 . . . . . 6 (∃𝑥𝜓 → ((∃𝑥𝜓𝑦 ∈ (V × V)) ↔ 𝑦 ∈ (V × V)))
31, 2syl 17 . . . . 5 (𝜑 → ((∃𝑥𝜓𝑦 ∈ (V × V)) ↔ 𝑦 ∈ (V × V)))
43bicomd 214 . . . 4 (𝜑 → (𝑦 ∈ (V × V) ↔ (∃𝑥𝜓𝑦 ∈ (V × V))))
54anbi1d 617 . . 3 (𝜑 → ((𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓𝑦𝑥)) ↔ ((∃𝑥𝜓𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓𝑦𝑥))))
6 elcnvintab 38402 . . 3 (𝑦 {𝑥𝜓} ↔ (𝑦 ∈ (V × V) ∧ ∀𝑥(𝜓𝑦𝑥)))
7 vex 3390 . . . 4 𝑦 ∈ V
8 vex 3390 . . . . . 6 𝑥 ∈ V
98cnvex 7337 . . . . 5 𝑥 ∈ V
10 relcnv 5707 . . . . . 6 Rel 𝑥
11 df-rel 5312 . . . . . 6 (Rel 𝑥𝑥 ⊆ (V × V))
1210, 11mpbi 221 . . . . 5 𝑥 ⊆ (V × V)
139, 12elmapintrab 38376 . . . 4 (𝑦 ∈ V → (𝑦 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)} ↔ ((∃𝑥𝜓𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓𝑦𝑥))))
147, 13ax-mp 5 . . 3 (𝑦 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)} ↔ ((∃𝑥𝜓𝑦 ∈ (V × V)) ∧ ∀𝑥(𝜓𝑦𝑥)))
155, 6, 143bitr4g 305 . 2 (𝜑 → (𝑦 {𝑥𝜓} ↔ 𝑦 {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)}))
1615eqrdv 2800 1 (𝜑 {𝑥𝜓} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2155  {cab 2788  {crab 3096  Vcvv 3387  wss 3763  𝒫 cpw 4345   cint 4662   × cxp 5303  ccnv 5304  Rel wrel 5310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-int 4663  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-iota 6058  df-fun 6097  df-fv 6103  df-1st 7392  df-2nd 7393
This theorem is referenced by:  clcnvlem  38424
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