Theorem cosseq 39143

Description: Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)

Assertion

Ref	Expression
cosseq	⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)

Proof of Theorem cosseq

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 5112	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑥 ↔ 𝑢𝐵𝑥))
2		breq 5112	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑦 ↔ 𝑢𝐵𝑦))
3	1, 2	anbi12d 643	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)))
4	3	exbidv 1951	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)))
5	4	opabbidv 5178	. 2 ⊢ (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)})
6		df-coss 39128	. 2 ⊢ ≀ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)}
7		df-coss 39128	. 2 ⊢ ≀ 𝐵 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)}
8	5, 6, 7	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   class class class wbr 5110  {copab 5174   ≀ ccoss 38810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-br 5111  df-opab 5175  df-coss 39128

This theorem is referenced by:  cosseqi  39144  cosseqd  39145  elfunsALTV  39404