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Theorem fliftcnv 6516
Description: Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2 ((𝜑𝑥𝑋) → 𝐴𝑅)
flift.3 ((𝜑𝑥𝑋) → 𝐵𝑆)
Assertion
Ref Expression
fliftcnv (𝜑𝐹 = ran (𝑥𝑋 ↦ ⟨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.
StepHypRef Expression
1 eqid 2626 . . . . 5 ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩) = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)
2 flift.3 . . . . 5 ((𝜑𝑥𝑋) → 𝐵𝑆)
3 flift.2 . . . . 5 ((𝜑𝑥𝑋) → 𝐴𝑅)
41, 2, 3fliftrel 6513 . . . 4 (𝜑 → ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅))
5 relxp 5193 . . . 4 Rel (𝑆 × 𝑅)
6 relss 5172 . . . 4 (ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)))
74, 5, 6mpisyl 21 . . 3 (𝜑 → Rel ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
8 relcnv 5466 . . 3 Rel 𝐹
97, 8jctil 559 . 2 (𝜑 → (Rel 𝐹 ∧ Rel ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)))
10 flift.1 . . . . . . 7 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
1110, 3, 2fliftel 6514 . . . . . 6 (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑦 = 𝐵)))
12 vex 3194 . . . . . . 7 𝑦 ∈ V
13 vex 3194 . . . . . . 7 𝑧 ∈ V
1412, 13brcnv 5270 . . . . . 6 (𝑦𝐹𝑧𝑧𝐹𝑦)
15 ancom 466 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐴) ↔ (𝑧 = 𝐴𝑦 = 𝐵))
1615rexbii 3039 . . . . . 6 (∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴) ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑦 = 𝐵))
1711, 14, 163bitr4g 303 . . . . 5 (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴)))
181, 2, 3fliftel 6514 . . . . 5 (𝜑 → (𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴)))
1917, 18bitr4d 271 . . . 4 (𝜑 → (𝑦𝐹𝑧𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧))
20 df-br 4619 . . . 4 (𝑦𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐹)
21 df-br 4619 . . . 4 (𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
2219, 20, 213bitr3g 302 . . 3 (𝜑 → (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)))
2322eqrelrdv2 5185 . 2 (((Rel 𝐹 ∧ Rel ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)) ∧ 𝜑) → 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
249, 23mpancom 702 1 (𝜑𝐹 = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wrex 2913  wss 3560  cop 4159   class class class wbr 4618  cmpt 4678   × cxp 5077  ccnv 5078  ran crn 5080  Rel wrel 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858
This theorem is referenced by:  pi1xfrcnvlem  22759
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