Theorem supeq3 7188

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 4090	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑥𝑅𝑦 ↔ 𝑥𝑆𝑦))
2	1	notbid 673	. . . . . 6 ⊢ (𝑅 = 𝑆 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑆𝑦))
3	2	ralbidv 2532	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦))
4		breq 4090	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥))
5		breq 4090	. . . . . . . 8 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑧 ↔ 𝑦𝑆𝑧))
6	5	rexbidv 2533	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (∃𝑧 ∈ 𝐴 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))
7	4, 6	imbi12d 234	. . . . . 6 ⊢ (𝑅 = 𝑆 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧) ↔ (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧)))
8	7	ralbidv 2532	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧)))
9	3, 8	anbi12d 473	. . . 4 ⊢ (𝑅 = 𝑆 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))))
10	9	rabbidv 2791	. . 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 3904	. 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 7182	. 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
13		df-sup 7182	. 2 ⊢ sup(𝐴, 𝐵, 𝑆) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))}
14	11, 12, 13	3eqtr4g 2289	1 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1397  ∀wral 2510  ∃wrex 2511  {crab 2514  ∪ cuni 3893   class class class wbr 4088  supcsup 7180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-uni 3894  df-br 4089  df-sup 7182

This theorem is referenced by:  infeq3  7213