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Theorem supeq2 9472
Description: Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
supeq2 (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))

Proof of Theorem supeq2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabeq 3443 . . . 4 (𝐵 = 𝐶 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
2 raleq 3319 . . . . . 6 (𝐵 = 𝐶 → (∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧) ↔ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)))
32anbi2d 629 . . . . 5 (𝐵 = 𝐶 → ((∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)) ↔ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))))
43rabbidv 3437 . . . 4 (𝐵 = 𝐶 → {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
51, 4eqtrd 2768 . . 3 (𝐵 = 𝐶 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
65unieqd 4921 . 2 (𝐵 = 𝐶 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
7 df-sup 9466 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
8 df-sup 9466 . 2 sup(𝐴, 𝐶, 𝑅) = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
96, 7, 83eqtr4g 2793 1 (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wral 3058  wrex 3067  {crab 3429   cuni 4908   class class class wbr 5148  supcsup 9464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909  df-sup 9466
This theorem is referenced by:  infeq2  9503
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