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Theorem supeq3 8396
Description: Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
Assertion
Ref Expression
supeq3 (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))

Proof of Theorem supeq3
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4687 . . . . . . 7 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
21notbid 307 . . . . . 6 (𝑅 = 𝑆 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑆𝑦))
32ralbidv 3015 . . . . 5 (𝑅 = 𝑆 → (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥𝑆𝑦))
4 breq 4687 . . . . . . 7 (𝑅 = 𝑆 → (𝑦𝑅𝑥𝑦𝑆𝑥))
5 breq 4687 . . . . . . . 8 (𝑅 = 𝑆 → (𝑦𝑅𝑧𝑦𝑆𝑧))
65rexbidv 3081 . . . . . . 7 (𝑅 = 𝑆 → (∃𝑧𝐴 𝑦𝑅𝑧 ↔ ∃𝑧𝐴 𝑦𝑆𝑧))
74, 6imbi12d 333 . . . . . 6 (𝑅 = 𝑆 → ((𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧) ↔ (𝑦𝑆𝑥 → ∃𝑧𝐴 𝑦𝑆𝑧)))
87ralbidv 3015 . . . . 5 (𝑅 = 𝑆 → (∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧) ↔ ∀𝑦𝐵 (𝑦𝑆𝑥 → ∃𝑧𝐴 𝑦𝑆𝑧)))
93, 8anbi12d 747 . . . 4 (𝑅 = 𝑆 → ((∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)) ↔ (∀𝑦𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦𝐵 (𝑦𝑆𝑥 → ∃𝑧𝐴 𝑦𝑆𝑧))))
109rabbidv 3220 . . 3 (𝑅 = 𝑆 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦𝐵 (𝑦𝑆𝑥 → ∃𝑧𝐴 𝑦𝑆𝑧))})
1110unieqd 4478 . 2 (𝑅 = 𝑆 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦𝐵 (𝑦𝑆𝑥 → ∃𝑧𝐴 𝑦𝑆𝑧))})
12 df-sup 8389 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
13 df-sup 8389 . 2 sup(𝐴, 𝐵, 𝑆) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦𝐵 (𝑦𝑆𝑥 → ∃𝑧𝐴 𝑦𝑆𝑧))}
1411, 12, 133eqtr4g 2710 1 (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wral 2941  wrex 2942  {crab 2945   cuni 4468   class class class wbr 4685  supcsup 8387
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-uni 4469  df-br 4686  df-sup 8389
This theorem is referenced by:  infeq3  8427
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