Theorem fcoinvbr 32807

Description: Binary relation for the equivalence relation from fcoinver 32806. (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 5840	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋(◡𝐹 ∘ 𝐹)𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
4	2, 3	bitrid 285	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
5	4	3adant1 1144	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
6		fvex 6882	. . . . 5 ⊢ (𝐹‘𝑋) ∈ V
7	6	eqvinc 3610	. . . 4 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧(𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)))
8		eqcom 2771	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = 𝑧)
9		eqcom 2771	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑌) ↔ (𝐹‘𝑌) = 𝑧)
10	8, 9	anbi12i 637	. . . . 5 ⊢ ((𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)) ↔ ((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
11	10	exbii 1870	. . . 4 ⊢ (∃𝑧(𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)) ↔ ∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
12	7, 11	bitri 277	. . 3 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
13		fnbrfvb 6919	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑧 ↔ 𝑋𝐹𝑧))
14	13	3adant3 1146	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑧 ↔ 𝑋𝐹𝑧))
15		fnbrfvb 6919	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = 𝑧 ↔ 𝑌𝐹𝑧))
16	15	3adant2 1145	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = 𝑧 ↔ 𝑌𝐹𝑧))
17	14, 16	anbi12d 641	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ (𝑋𝐹𝑧 ∧ 𝑌𝐹𝑧)))
18		vex 3460	. . . . . . . 8 ⊢ 𝑧 ∈ V
19		brcnvg 5853	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑌 ∈ 𝐴) → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
20	18, 19	mpan 700	. . . . . . 7 ⊢ (𝑌 ∈ 𝐴 → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
21	20	3ad2ant3 1149	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
22	21	anbi2d 639	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌) ↔ (𝑋𝐹𝑧 ∧ 𝑌𝐹𝑧)))
23	17, 22	bitr4d 284	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ (𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
24	23	exbidv 1943	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
25	12, 24	bitrid 285	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
26	5, 25	bitr4d 284	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562  ∃wex 1801   ∈ wcel 2144  Vcvv 3456   class class class wbr 5102  ^◡ccnv 5648   ∘ ccom 5653   Fn wfn 6518  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531

This theorem is referenced by:  qtophaus  34135