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Theorem supeq2 8904
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 3488 . . . 4 (𝐵 = 𝐶 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
2 raleq 3410 . . . . . 6 (𝐵 = 𝐶 → (∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧) ↔ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)))
32anbi2d 628 . . . . 5 (𝐵 = 𝐶 → ((∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)) ↔ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))))
43rabbidv 3485 . . . 4 (𝐵 = 𝐶 → {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
51, 4eqtrd 2860 . . 3 (𝐵 = 𝐶 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
65unieqd 4846 . 2 (𝐵 = 𝐶 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))} = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))})
7 df-sup 8898 . 2 sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
8 df-sup 8898 . 2 sup(𝐴, 𝐶, 𝑅) = {𝑥𝐶 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐶 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
96, 7, 83eqtr4g 2885 1 (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wral 3142  wrex 3143  {crab 3146   cuni 4836   class class class wbr 5062  supcsup 8896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-ral 3147  df-rex 3148  df-rab 3151  df-uni 4837  df-sup 8898
This theorem is referenced by:  infeq2  8935
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