Theorem fliftcnv 7269

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 2737	. . . . 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 7266	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅))
5		relxp 5652	. . . 4 ⊢ Rel (𝑆 × 𝑅)
6		relss 5741	. . . 4 ⊢ (ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
7	4, 5, 6	mpisyl 21	. . 3 ⊢ (𝜑 → Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
8		relcnv 6073	. . 3 ⊢ Rel ◡𝐹
9	7, 8	jctil 519	. 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 7267	. . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)))
12		vex 3446	. . . . . . 7 ⊢ 𝑦 ∈ V
13		vex 3446	. . . . . . 7 ⊢ 𝑧 ∈ V
14	12, 13	brcnv 5841	. . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦)
15		ancom 460	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
16	15	rexbii 3085	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
17	11, 14, 16	3bitr4g 314	. . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
18	1, 2, 3	fliftel 7267	. . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
19	17, 18	bitr4d 282	. . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧))
20		df-br 5101	. . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ◡𝐹)
21		df-br 5101	. . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
22	19, 20, 21	3bitr3g 313	. . 3 ⊢ (𝜑 → (⟨𝑦, 𝑧⟩ ∈ ◡𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
23	22	eqrelrdv2 5754	. 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
24	9, 23	mpancom 689	1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181   × cxp 5632  ^◡ccnv 5633  ran crn 5635  Rel wrel 5639

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6504  df-fn 6505  df-f 6506

This theorem is referenced by:  pi1xfrcnvlem  25029