Theorem fcoinvbr 32627

Description: Binary relation for the equivalence relation from fcoinver 32626. (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 5172	. . . 4 ⊢ (𝑋 ∼ 𝑌 ↔ 𝑋(◡𝐹 ∘ 𝐹)𝑌)
3		brcog 5891	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋(◡𝐹 ∘ 𝐹)𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
4	2, 3	bitrid 283	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
5	4	3adant1 1130	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
6		fvex 6933	. . . . 5 ⊢ (𝐹‘𝑋) ∈ V
7	6	eqvinc 3662	. . . 4 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧(𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)))
8		eqcom 2747	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = 𝑧)
9		eqcom 2747	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑌) ↔ (𝐹‘𝑌) = 𝑧)
10	8, 9	anbi12i 627	. . . . 5 ⊢ ((𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)) ↔ ((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
11	10	exbii 1846	. . . 4 ⊢ (∃𝑧(𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)) ↔ ∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
12	7, 11	bitri 275	. . 3 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
13		fnbrfvb 6973	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑧 ↔ 𝑋𝐹𝑧))
14	13	3adant3 1132	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑧 ↔ 𝑋𝐹𝑧))
15		fnbrfvb 6973	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = 𝑧 ↔ 𝑌𝐹𝑧))
16	15	3adant2 1131	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = 𝑧 ↔ 𝑌𝐹𝑧))
17	14, 16	anbi12d 631	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ (𝑋𝐹𝑧 ∧ 𝑌𝐹𝑧)))
18		vex 3492	. . . . . . . 8 ⊢ 𝑧 ∈ V
19		brcnvg 5904	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑌 ∈ 𝐴) → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
20	18, 19	mpan 689	. . . . . . 7 ⊢ (𝑌 ∈ 𝐴 → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
21	20	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
22	21	anbi2d 629	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌) ↔ (𝑋𝐹𝑧 ∧ 𝑌𝐹𝑧)))
23	17, 22	bitr4d 282	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ (𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
24	23	exbidv 1920	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
25	12, 24	bitrid 283	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
26	5, 25	bitr4d 282	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  ^◡ccnv 5699   ∘ ccom 5704   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  qtophaus  33782