Theorem supeq3 6827

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 3895	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑥𝑅𝑦 ↔ 𝑥𝑆𝑦))
2	1	notbid 639	. . . . . 6 ⊢ (𝑅 = 𝑆 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑆𝑦))
3	2	ralbidv 2409	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦))
4		breq 3895	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥))
5		breq 3895	. . . . . . . 8 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑧 ↔ 𝑦𝑆𝑧))
6	5	rexbidv 2410	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (∃𝑧 ∈ 𝐴 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))
7	4, 6	imbi12d 233	. . . . . 6 ⊢ (𝑅 = 𝑆 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧) ↔ (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧)))
8	7	ralbidv 2409	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧)))
9	3, 8	anbi12d 462	. . . 4 ⊢ (𝑅 = 𝑆 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))))
10	9	rabbidv 2644	. . 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 3711	. 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 6821	. 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
13		df-sup 6821	. 2 ⊢ sup(𝐴, 𝐵, 𝑆) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))}
14	11, 12, 13	3eqtr4g 2170	1 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1312  ∀wral 2388  ∃wrex 2389  {crab 2392  ∪ cuni 3700   class class class wbr 3893  supcsup 6819

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 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-uni 3701  df-br 3894  df-sup 6821

This theorem is referenced by:  infeq3  6852