Theorem supeq123d 8709

Description: Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

Ref	Expression
supeq123d.a	⊢ (𝜑 → 𝐴 = 𝐷)
supeq123d.b	⊢ (𝜑 → 𝐵 = 𝐸)
supeq123d.c	⊢ (𝜑 → 𝐶 = 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.

Step	Hyp	Ref	Expression
1		supeq123d.b	. . . 4 ⊢ (𝜑 → 𝐵 = 𝐸)
2		supeq123d.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐷)
3		supeq123d.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = 𝐹)
4	3	breqd 4940	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐶𝑦 ↔ 𝑥𝐹𝑦))
5	4	notbid 310	. . . . . 6 ⊢ (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦))
6	2, 5	raleqbidv 3341	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦))
7	3	breqd 4940	. . . . . . 7 ⊢ (𝜑 → (𝑦𝐶𝑥 ↔ 𝑦𝐹𝑥))
8	3	breqd 4940	. . . . . . . 8 ⊢ (𝜑 → (𝑦𝐶𝑧 ↔ 𝑦𝐹𝑧))
9	2, 8	rexeqbidv 3342	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ 𝐴 𝑦𝐶𝑧 ↔ ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))
10	7, 9	imbi12d 337	. . . . . 6 ⊢ (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)))
11	1, 10	raleqbidv 3341	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)))
12	6, 11	anbi12d 621	. . . 4 ⊢ (𝜑 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))))
13	1, 12	rabeqbidv 3408	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧))} = {𝑥 ∈ 𝐸 ∣ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))})
14	13	unieqd 4722	. 2 ⊢ (𝜑 → ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧))} = ∪ {𝑥 ∈ 𝐸 ∣ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))})
15		df-sup 8701	. 2 ⊢ sup(𝐴, 𝐵, 𝐶) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧))}
16		df-sup 8701	. 2 ⊢ sup(𝐷, 𝐸, 𝐹) = ∪ {𝑥 ∈ 𝐸 ∣ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))}
17	14, 15, 16	3eqtr4g 2839	1 ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   = wceq 1507  ∀wral 3088  ∃wrex 3089  {crab 3092  ∪ cuni 4712   class class class wbr 4929  supcsup 8699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-ral 3093  df-rex 3094  df-rab 3097  df-uni 4713  df-br 4930  df-sup 8701

This theorem is referenced by:  infeq123d  8740  wlimeq12  32633  aomclem8  39063