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Theorem crefeq 32466
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 4171 . . . . . 6 (𝐴 = 𝐵 → (𝒫 𝑗𝐴) = (𝒫 𝑗𝐵))
21rexeqdv 3317 . . . . 5 (𝐴 = 𝐵 → (∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦))
32imbi2d 341 . . . 4 (𝐴 = 𝐵 → (( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ ( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)))
43ralbidv 3175 . . 3 (𝐴 = 𝐵 → (∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)))
54rabbidv 3418 . 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 32464 . 2 CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)}
7 df-cref 32464 . 2 CovHasRef𝐵 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐵)𝑧Ref𝑦)}
85, 6, 73eqtr4g 2802 1 (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wral 3065  wrex 3074  {crab 3410  cin 3914  𝒫 cpw 4565   cuni 4870   class class class wbr 5110  Topctop 22258  Refcref 22869  CovHasRefccref 32463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-in 3922  df-cref 32464
This theorem is referenced by:  ispcmp  32478
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