Theorem supeq3 6967

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 3991	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑥𝑅𝑦 ↔ 𝑥𝑆𝑦))
2	1	notbid 662	. . . . . 6 ⊢ (𝑅 = 𝑆 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑆𝑦))
3	2	ralbidv 2470	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦))
4		breq 3991	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑥 ↔ 𝑦𝑆𝑥))
5		breq 3991	. . . . . . . 8 ⊢ (𝑅 = 𝑆 → (𝑦𝑅𝑧 ↔ 𝑦𝑆𝑧))
6	5	rexbidv 2471	. . . . . . 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 2470	. . . . 5 ⊢ (𝑅 = 𝑆 → (∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧)))
9	3, 8	anbi12d 470	. . . 4 ⊢ (𝑅 = 𝑆 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))))
10	9	rabbidv 2719	. . 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 3807	. 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 6961	. 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
13		df-sup 6961	. 2 ⊢ sup(𝐴, 𝐵, 𝑆) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑆𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑆𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑆𝑧))}
14	11, 12, 13	3eqtr4g 2228	1 ⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1348  ∀wral 2448  ∃wrex 2449  {crab 2452  ∪ cuni 3796   class class class wbr 3989  supcsup 6959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-uni 3797  df-br 3990  df-sup 6961

This theorem is referenced by:  infeq3  6992