Theorem supeq123d 8356

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 4664	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐶𝑦 ↔ 𝑥𝐹𝑦))
5	4	notbid 308	. . . . . 6 ⊢ (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦))
6	2, 5	raleqbidv 3152	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦))
7	3	breqd 4664	. . . . . . 7 ⊢ (𝜑 → (𝑦𝐶𝑥 ↔ 𝑦𝐹𝑥))
8	3	breqd 4664	. . . . . . . 8 ⊢ (𝜑 → (𝑦𝐶𝑧 ↔ 𝑦𝐹𝑧))
9	2, 8	rexeqbidv 3153	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ 𝐴 𝑦𝐶𝑧 ↔ ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))
10	7, 9	imbi12d 334	. . . . . 6 ⊢ (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)))
11	1, 10	raleqbidv 3152	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)))
12	6, 11	anbi12d 747	. . . 4 ⊢ (𝜑 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))))
13	1, 12	rabeqbidv 3195	. . 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 4446	. 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 8348	. 2 ⊢ sup(𝐴, 𝐵, 𝐶) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧))}
16		df-sup 8348	. 2 ⊢ sup(𝐷, 𝐸, 𝐹) = ∪ {𝑥 ∈ 𝐸 ∣ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))}
17	14, 15, 16	3eqtr4g 2681	1 ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483  ∀wral 2912  ∃wrex 2913  {crab 2916  ∪ cuni 4436   class class class wbr 4653  supcsup 8346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-uni 4437  df-br 4654  df-sup 8348

This theorem is referenced by:  infeq123d  8387  wlimeq12  31765  aomclem8  37631