Theorem supeq123d 9066

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 5064	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐶𝑦 ↔ 𝑥𝐹𝑦))
5	4	notbid 321	. . . . . 6 ⊢ (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦))
6	2, 5	raleqbidv 3313	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦))
7	3	breqd 5064	. . . . . . 7 ⊢ (𝜑 → (𝑦𝐶𝑥 ↔ 𝑦𝐹𝑥))
8	3	breqd 5064	. . . . . . . 8 ⊢ (𝜑 → (𝑦𝐶𝑧 ↔ 𝑦𝐹𝑧))
9	2, 8	rexeqbidv 3314	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ 𝐴 𝑦𝐶𝑧 ↔ ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))
10	7, 9	imbi12d 348	. . . . . 6 ⊢ (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)))
11	1, 10	raleqbidv 3313	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)))
12	6, 11	anbi12d 634	. . . 4 ⊢ (𝜑 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))))
13	1, 12	rabeqbidv 3396	. . 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 4833	. 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 9058	. 2 ⊢ sup(𝐴, 𝐵, 𝐶) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧))}
16		df-sup 9058	. 2 ⊢ sup(𝐷, 𝐸, 𝐹) = ∪ {𝑥 ∈ 𝐸 ∣ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))}
17	14, 15, 16	3eqtr4g 2803	1 ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543  ∀wral 3061  ∃wrex 3062  {crab 3065  ∪ cuni 4819   class class class wbr 5053  supcsup 9056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883  df-uni 4820  df-br 5054  df-sup 9058

This theorem is referenced by:  infeq123d  9097  wlimeq12  33550  aomclem8  40589