Theorem fliftcnv 5838

Description: Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
flift.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)

Assertion

Ref	Expression
fliftcnv	⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))

Distinct variable groups:   𝑥,𝑅   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftcnv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2193	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)
2		flift.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
3		flift.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
4	1, 2, 3	fliftrel 5835	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅))
5		relxp 4768	. . . 4 ⊢ Rel (𝑆 × 𝑅)
6		relss 4746	. . . 4 ⊢ (ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
7	4, 5, 6	mpisyl 1457	. . 3 ⊢ (𝜑 → Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
8		relcnv 5043	. . 3 ⊢ Rel ◡𝐹
9	7, 8	jctil 312	. 2 ⊢ (𝜑 → (Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
10		flift.1	. . . . . . 7 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
11	10, 3, 2	fliftel 5836	. . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)))
12		vex 2763	. . . . . . 7 ⊢ 𝑦 ∈ V
13		vex 2763	. . . . . . 7 ⊢ 𝑧 ∈ V
14	12, 13	brcnv 4845	. . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦)
15		ancom 266	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
16	15	rexbii 2501	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
17	11, 14, 16	3bitr4g 223	. . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
18	1, 2, 3	fliftel 5836	. . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
19	17, 18	bitr4d 191	. . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧))
20		df-br 4030	. . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ◡𝐹)
21		df-br 4030	. . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
22	19, 20, 21	3bitr3g 222	. . 3 ⊢ (𝜑 → (⟨𝑦, 𝑧⟩ ∈ ◡𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
23	22	eqrelrdv2 4758	. 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
24	9, 23	mpancom 422	1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∃wrex 2473   ⊆ wss 3153  ⟨cop 3621   class class class wbr 4029   ↦ cmpt 4090   × cxp 4657  ^◡ccnv 4658  ran crn 4660  Rel wrel 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262

This theorem is referenced by: (None)