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Theorem supeq2 8899
 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 3431 . . . 4 (𝐵 = 𝐶 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
2 raleq 3358 . . . . . 6 (𝐵 = 𝐶 → (∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧) ↔ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)))
32anbi2d 631 . . . . 5 (𝐵 = 𝐶 → ((∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)) ↔ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))))
43rabbidv 3427 . . . 4 (𝐵 = 𝐶 → {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
51, 4eqtrd 2833 . . 3 (𝐵 = 𝐶 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
65unieqd 4815 . 2 (𝐵 = 𝐶 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
7 df-sup 8893 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
8 df-sup 8893 . 2 sup(𝐴, 𝐶, 𝑅) = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
96, 7, 83eqtr4g 2858 1 (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ∀wral 3106  ∃wrex 3107  {crab 3110  ∪ cuni 4801   class class class wbr 5031  supcsup 8891 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-uni 4802  df-sup 8893 This theorem is referenced by:  infeq2  8930
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