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Theorem crefdf 31011
Description: A formulation of crefi 31010 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Hypotheses
Ref Expression
crefi.x 𝑋 = 𝐽
crefdf.b 𝐵 = CovHasRef𝐴
crefdf.p (𝑧𝐴𝜑)
Assertion
Ref Expression
crefdf ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐽   𝑧,𝐶
Allowed substitution hints:   𝜑(𝑧)   𝐵(𝑧)   𝑋(𝑧)

Proof of Theorem crefdf
StepHypRef Expression
1 crefdf.b . . . 4 𝐵 = CovHasRef𝐴
21eleq2i 2901 . . 3 (𝐽𝐵𝐽 ∈ CovHasRef𝐴)
3 crefi.x . . . 4 𝑋 = 𝐽
43crefi 31010 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
52, 4syl3an1b 1395 . 2 ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
6 elin 4166 . . . . . 6 (𝑧 ∈ (𝒫 𝐽𝐴) ↔ (𝑧 ∈ 𝒫 𝐽𝑧𝐴))
7 crefdf.p . . . . . . 7 (𝑧𝐴𝜑)
87anim2i 616 . . . . . 6 ((𝑧 ∈ 𝒫 𝐽𝑧𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
96, 8sylbi 218 . . . . 5 (𝑧 ∈ (𝒫 𝐽𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
109anim1i 614 . . . 4 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶))
11 anass 469 . . . 4 (((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1210, 11sylib 219 . . 3 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1312reximi2 3241 . 2 (∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶 → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))
145, 13syl 17 1 ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  Refcref 22038  CovHasRefccref 31005
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
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-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  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-br 5058  df-cref 31006
This theorem is referenced by:  cmpfiref  31014  ldlfcntref  31017
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