Theorem fliftcnv 7066

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 2823	. . . . 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 7063	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅))
5		relxp 5575	. . . 4 ⊢ Rel (𝑆 × 𝑅)
6		relss 5658	. . . 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 5969	. . 3 ⊢ Rel ◡𝐹
9	7, 8	jctil 522	. 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 7064	. . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)))
12		vex 3499	. . . . . . 7 ⊢ 𝑦 ∈ V
13		vex 3499	. . . . . . 7 ⊢ 𝑧 ∈ V
14	12, 13	brcnv 5755	. . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦)
15		ancom 463	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
16	15	rexbii 3249	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
17	11, 14, 16	3bitr4g 316	. . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
18	1, 2, 3	fliftel 7064	. . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
19	17, 18	bitr4d 284	. . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧))
20		df-br 5069	. . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ◡𝐹)
21		df-br 5069	. . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
22	19, 20, 21	3bitr3g 315	. . 3 ⊢ (𝜑 → (⟨𝑦, 𝑧⟩ ∈ ◡𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
23	22	eqrelrdv2 5670	. 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
24	9, 23	mpancom 686	1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3141   ⊆ wss 3938  ⟨cop 4575   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  ^◡ccnv 5556  ran crn 5558  Rel wrel 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365

This theorem is referenced by:  pi1xfrcnvlem  23662