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Theorem supeq2 6869
 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 2673 . . . 4 (𝐵 = 𝐶 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
2 raleq 2624 . . . . . 6 (𝐵 = 𝐶 → (∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧) ↔ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)))
32anbi2d 459 . . . . 5 (𝐵 = 𝐶 → ((∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)) ↔ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))))
43rabbidv 2670 . . . 4 (𝐵 = 𝐶 → {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
51, 4eqtrd 2170 . . 3 (𝐵 = 𝐶 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
65unieqd 3742 . 2 (𝐵 = 𝐶 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
7 df-sup 6864 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
8 df-sup 6864 . 2 sup(𝐴, 𝐶, 𝑅) = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
96, 7, 83eqtr4g 2195 1 (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331  ∀wral 2414  ∃wrex 2415  {crab 2418  ∪ cuni 3731   class class class wbr 3924  supcsup 6862 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-uni 3732  df-sup 6864 This theorem is referenced by:  infeq2  6894
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