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Theorem crefeq 30040
Description: Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Assertion
Ref Expression
crefeq (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)

Proof of Theorem crefeq
Dummy variables 𝑗 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ineq2 3841 . . . . . 6 (𝐴 = 𝐵 → (𝒫 𝑗𝐴) = (𝒫 𝑗𝐵))
21rexeqdv 3175 . . . . 5 (𝐴 = 𝐵 → (∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦))
32imbi2d 329 . . . 4 (𝐴 = 𝐵 → (( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ ( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)))
43ralbidv 3015 . . 3 (𝐴 = 𝐵 → (∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)))
54rabbidv 3220 . 2 (𝐴 = 𝐵 → {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)} = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)})
6 df-cref 30038 . 2 CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)}
7 df-cref 30038 . 2 CovHasRef𝐵 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)}
85, 6, 73eqtr4g 2710 1 (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wral 2941  wrex 2942  {crab 2945  cin 3606  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  Topctop 20746  Refcref 21353  CovHasRefccref 30037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-in 3614  df-cref 30038
This theorem is referenced by:  ispcmp  30052
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