Theorem supeq3 9333

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.

Step	Hyp	Ref	Expression
1		breq 5091	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑥𝑅𝑦 ↔ 𝑥𝑆𝑦))
2	1	notbid 318	. . . . . 6 ⊢ (𝑅 = 𝑆 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑆𝑦))
3	2	ralbidv 3155	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦))
4		breq 5091	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥))
5		breq 5091	. . . . . . . 8 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑧 ↔ 𝑦𝑆𝑧))
6	5	rexbidv 3156	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (∃𝑧 ∈ 𝐴 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))
7	4, 6	imbi12d 344	. . . . . 6 ⊢ (𝑅 = 𝑆 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧) ↔ (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧)))
8	7	ralbidv 3155	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧)))
9	3, 8	anbi12d 632	. . . 4 ⊢ (𝑅 = 𝑆 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))))
10	9	rabbidv 3402	. . 3 ⊢ (𝑅 = 𝑆 → {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))} = {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))})
11	10	unieqd 4869	. 2 ⊢ (𝑅 = 𝑆 → ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))} = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))})
12		df-sup 9326	. 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
13		df-sup 9326	. 2 ⊢ sup(𝐴, 𝐵, 𝑆) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))}
14	11, 12, 13	3eqtr4g 2791	1 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541  ∀wral 3047  ∃wrex 3056  {crab 3395  ∪ cuni 4856   class class class wbr 5089  supcsup 9324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-ss 3914  df-uni 4857  df-br 5090  df-sup 9326

This theorem is referenced by:  infeq3  9365