MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supeq123d Structured version   Visualization version   GIF version

Theorem supeq123d 9066
Description: Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
supeq123d.a (𝜑𝐴 = 𝐷)
supeq123d.b (𝜑𝐵 = 𝐸)
supeq123d.c (𝜑𝐶 = 𝐹)
Assertion
Ref Expression
supeq123d (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))

Proof of Theorem supeq123d
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supeq123d.b . . . 4 (𝜑𝐵 = 𝐸)
2 supeq123d.a . . . . . 6 (𝜑𝐴 = 𝐷)
3 supeq123d.c . . . . . . . 8 (𝜑𝐶 = 𝐹)
43breqd 5064 . . . . . . 7 (𝜑 → (𝑥𝐶𝑦𝑥𝐹𝑦))
54notbid 321 . . . . . 6 (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦))
62, 5raleqbidv 3313 . . . . 5 (𝜑 → (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦𝐷 ¬ 𝑥𝐹𝑦))
73breqd 5064 . . . . . . 7 (𝜑 → (𝑦𝐶𝑥𝑦𝐹𝑥))
83breqd 5064 . . . . . . . 8 (𝜑 → (𝑦𝐶𝑧𝑦𝐹𝑧))
92, 8rexeqbidv 3314 . . . . . . 7 (𝜑 → (∃𝑧𝐴 𝑦𝐶𝑧 ↔ ∃𝑧𝐷 𝑦𝐹𝑧))
107, 9imbi12d 348 . . . . . 6 (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
111, 10raleqbidv 3313 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
126, 11anbi12d 634 . . . 4 (𝜑 → ((∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))))
131, 12rabeqbidv 3396 . . 3 (𝜑 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
1413unieqd 4833 . 2 (𝜑 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
15 df-sup 9058 . 2 sup(𝐴, 𝐵, 𝐶) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))}
16 df-sup 9058 . 2 sup(𝐷, 𝐸, 𝐹) = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))}
1714, 15, 163eqtr4g 2803 1 (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wral 3061  wrex 3062  {crab 3065   cuni 4819   class class class wbr 5053  supcsup 9056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883  df-uni 4820  df-br 5054  df-sup 9058
This theorem is referenced by:  infeq123d  9097  wlimeq12  33550  aomclem8  40589
  Copyright terms: Public domain W3C validator