Theorem supeq123d 6830

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 3906	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐶𝑦 ↔ 𝑥𝐹𝑦))
5	4	notbid 639	. . . . . 6 ⊢ (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦))
6	2, 5	raleqbidv 2612	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦))
7	3	breqd 3906	. . . . . . 7 ⊢ (𝜑 → (𝑦𝐶𝑥 ↔ 𝑦𝐹𝑥))
8	3	breqd 3906	. . . . . . . 8 ⊢ (𝜑 → (𝑦𝐶𝑧 ↔ 𝑦𝐹𝑧))
9	2, 8	rexeqbidv 2613	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ 𝐴 𝑦𝐶𝑧 ↔ ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))
10	7, 9	imbi12d 233	. . . . . 6 ⊢ (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)))
11	1, 10	raleqbidv 2612	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)))
12	6, 11	anbi12d 462	. . . 4 ⊢ (𝜑 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))))
13	1, 12	rabeqbidv 2652	. . 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 3713	. 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 6823	. 2 ⊢ sup(𝐴, 𝐵, 𝐶) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧))}
16		df-sup 6823	. 2 ⊢ sup(𝐷, 𝐸, 𝐹) = ∪ {𝑥 ∈ 𝐸 ∣ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))}
17	14, 15, 16	3eqtr4g 2172	1 ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1314  ∀wral 2390  ∃wrex 2391  {crab 2394  ∪ cuni 3702   class class class wbr 3895  supcsup 6821

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-uni 3703  df-br 3896  df-sup 6823

This theorem is referenced by:  infeq123d  6855