Theorem fcoinvbr 30371

Description: Binary relation for the equivalence relation from fcoinver 30370. (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 5036	. . . 4 ⊢ (𝑋 ∼ 𝑌 ↔ 𝑋(◡𝐹 ∘ 𝐹)𝑌)
3		brcog 5701	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋(◡𝐹 ∘ 𝐹)𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
4	2, 3	syl5bb 286	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
5	4	3adant1 1127	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 ∼ 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
6		fvex 6658	. . . . 5 ⊢ (𝐹‘𝑋) ∈ V
7	6	eqvinc 3590	. . . 4 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) ↔ ∃𝑧(𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)))
8		eqcom 2805	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = 𝑧)
9		eqcom 2805	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑌) ↔ (𝐹‘𝑌) = 𝑧)
10	8, 9	anbi12i 629	. . . . 5 ⊢ ((𝑧 = (𝐹‘𝑋) ∧ 𝑧 = (𝐹‘𝑌)) ↔ ((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧))
11	10	exbii 1849	. . . 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 6693	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑧 ↔ 𝑋𝐹𝑧))
14	13	3adant3 1129	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑧 ↔ 𝑋𝐹𝑧))
15		fnbrfvb 6693	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = 𝑧 ↔ 𝑌𝐹𝑧))
16	15	3adant2 1128	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = 𝑧 ↔ 𝑌𝐹𝑧))
17	14, 16	anbi12d 633	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ (𝑋𝐹𝑧 ∧ 𝑌𝐹𝑧)))
18		vex 3444	. . . . . . . 8 ⊢ 𝑧 ∈ V
19		brcnvg 5714	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑌 ∈ 𝐴) → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
20	18, 19	mpan 689	. . . . . . 7 ⊢ (𝑌 ∈ 𝐴 → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
21	20	3ad2ant3 1132	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑧◡𝐹𝑌 ↔ 𝑌𝐹𝑧))
22	21	anbi2d 631	. . . . 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 1922	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (∃𝑧((𝐹‘𝑋) = 𝑧 ∧ (𝐹‘𝑌) = 𝑧) ↔ ∃𝑧(𝑋𝐹𝑧 ∧ 𝑧◡𝐹𝑌)))
25	12, 24	syl5bb 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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   class class class wbr 5030  ^◡ccnv 5518   ∘ ccom 5523   Fn wfn 6319  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332

This theorem is referenced by:  qtophaus  31189