Theorem fliftcnv 7175

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 2739	. . . . 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 7172	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅))
5		relxp 5606	. . . 4 ⊢ Rel (𝑆 × 𝑅)
6		relss 5690	. . . 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 6009	. . 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 7173	. . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)))
12		vex 3434	. . . . . . 7 ⊢ 𝑦 ∈ V
13		vex 3434	. . . . . . 7 ⊢ 𝑧 ∈ V
14	12, 13	brcnv 5788	. . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦)
15		ancom 460	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
16	15	rexbii 3179	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
17	11, 14, 16	3bitr4g 313	. . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
18	1, 2, 3	fliftel 7173	. . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
19	17, 18	bitr4d 281	. . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧))
20		df-br 5079	. . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ◡𝐹)
21		df-br 5079	. . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
22	19, 20, 21	3bitr3g 312	. . 3 ⊢ (𝜑 → (⟨𝑦, 𝑧⟩ ∈ ◡𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
23	22	eqrelrdv2 5702	. 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
24	9, 23	mpancom 684	1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ∃wrex 3066   ⊆ wss 3891  ⟨cop 4572   class class class wbr 5078   ↦ cmpt 5161   × cxp 5586  ^◡ccnv 5587  ran crn 5589  Rel wrel 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-fun 6432  df-fn 6433  df-f 6434

This theorem is referenced by:  pi1xfrcnvlem  24200