Theorem supeq1 9474

Description: Equality theorem for supremum. (Contributed by NM, 22-May-1999.)

Assertion

Ref	Expression
supeq1	⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))

Proof of Theorem supeq1

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3320	. . . . 5 ⊢ (𝐵 = 𝐶 → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦))
2		rexeq 3319	. . . . . . 7 ⊢ (𝐵 = 𝐶 → (∃𝑧 ∈ 𝐵 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))
3	2	imbi2d 343	. . . . . 6 ⊢ (𝐵 = 𝐶 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))
4	3	ralbidv 3189	. . . . 5 ⊢ (𝐵 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))
5	1, 4	anbi12d 644	. . . 4 ⊢ (𝐵 = 𝐶 → ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ↔ (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))))
6	5	rabbidv 3423	. . 3 ⊢ (𝐵 = 𝐶 → {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))})
7	6	unieqd 4885	. 2 ⊢ (𝐵 = 𝐶 → ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))})
8		df-sup 9471	. 2 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
9		df-sup 9471	. 2 ⊢ sup(𝐶, 𝐴, 𝑅) = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))}
10	7, 8, 9	3eqtr4g 2823	1 ⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∀wral 3080  ∃wrex 3090  {crab 3416  ∪ cuni 4872   class class class wbr 5108  supcsup 9469

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-uni 4873  df-sup 9471

This theorem is used by:  supeq1d  9475  supeq1i  9476  infeq1  9506  bndth  25334  ioorval  25950  uniioombllem6  25964  mdegcl  26442  limexissupab  44546  suplesup  46624  supminfxr  46747  prproropf1olem2  48859  prproropf1olem3  48860  prproropf1olem4  48861  prproropf1o  48862  prproropreud  48864