Theorem fcoinvbr 33199

Description: Binary relation for the equivalence relation from fcoinver 33198. (Contributed by Thierry Arnoux, 3-Jan-2020.)

Hypothesis

Ref	Expression
fcoinvbr.e	⊢ ∼ = (◡𝐹 ∘ 𝐹)

Assertion

Ref	Expression
fcoinvbr	⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))

Proof of Theorem fcoinvbr

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fcoinvbr.e	. . . . 5 ⊢ ∼ = (◡𝐹 ∘ 𝐹)
2	1	breqi 5108	. . . 4 ⊢ (𝑋 ∼ 𝑌 ↔ 𝑋(◡𝐹 ∘ 𝐹)𝑌)
3		brcog 5843	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋(◡𝐹 ∘ 𝐹)𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
4	2, 3	bitrid 286	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
5	4	3adant1 1148	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
6		fvex 6898	. . . . 5 ⊢ (𝐹‘𝑋) ∈ V
7	6	eqvinc 3602	. . . 4 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧(𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)))
8		eqcom 2767	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = 𝑧)
9		eqcom 2767	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑌) ↔ (𝐹‘𝑌) = 𝑧)
10	8, 9	anbi12i 640	. . . . 5 ⊢ ((𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)) ↔ ((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
11	10	exbii 1880	. . . 4 ⊢ (∃𝑧(𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)) ↔ ∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
12	7, 11	bitri 278	. . 3 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
13		fnbrfvb 6935	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑧 ↔ 𝑋𝐹𝑧))
14	13	3adant3 1150	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑧 ↔ 𝑋𝐹𝑧))
15		fnbrfvb 6935	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = 𝑧 ↔ 𝑌𝐹𝑧))
16	15	3adant2 1149	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = 𝑧 ↔ 𝑌𝐹𝑧))
17	14, 16	anbi12d 644	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ (𝑋𝐹𝑧 ∧ 𝑌𝐹𝑧)))
18		vex 3454	. . . . . . . 8 ⊢ 𝑧 ∈ V
19		brcnvg 5856	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑌 ∈ 𝐴) → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
20	18, 19	mpan 703	. . . . . . 7 ⊢ (𝑌 ∈ 𝐴 → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
21	20	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
22	21	anbi2d 642	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌) ↔ (𝑋𝐹𝑧 ∧ 𝑌𝐹𝑧)))
23	17, 22	bitr4d 285	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ (𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
24	23	exbidv 1953	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
25	12, 24	bitrid 286	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
26	5, 25	bitr4d 285	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450   class class class wbr 5102  ^◡ccnv 5649   ∘ ccom 5654   Fn wfn 6533  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by:  qtophaus  34468