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Theorem fliftcnv 7053
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 2818 . . . . 5 ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩) = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)
2 flift.3 . . . . 5 ((𝜑𝑥𝑋) → 𝐵𝑆)
3 flift.2 . . . . 5 ((𝜑𝑥𝑋) → 𝐴𝑅)
41, 2, 3fliftrel 7050 . . . 4 (𝜑 → ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅))
5 relxp 5566 . . . 4 Rel (𝑆 × 𝑅)
6 relss 5649 . . . 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 5960 . . 3 Rel 𝐹
97, 8jctil 520 . 2 (𝜑 → (Rel 𝐹 ∧ Rel ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)))
10 flift.1 . . . . . . 7 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
1110, 3, 2fliftel 7051 . . . . . 6 (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑦 = 𝐵)))
12 vex 3495 . . . . . . 7 𝑦 ∈ V
13 vex 3495 . . . . . . 7 𝑧 ∈ V
1412, 13brcnv 5746 . . . . . 6 (𝑦𝐹𝑧𝑧𝐹𝑦)
15 ancom 461 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐴) ↔ (𝑧 = 𝐴𝑦 = 𝐵))
1615rexbii 3244 . . . . . 6 (∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴) ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑦 = 𝐵))
1711, 14, 163bitr4g 315 . . . . 5 (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴)))
181, 2, 3fliftel 7051 . . . . 5 (𝜑 → (𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴)))
1917, 18bitr4d 283 . . . 4 (𝜑 → (𝑦𝐹𝑧𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧))
20 df-br 5058 . . . 4 (𝑦𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐹)
21 df-br 5058 . . . 4 (𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
2219, 20, 213bitr3g 314 . . 3 (𝜑 → (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)))
2322eqrelrdv2 5661 . 2 (((Rel 𝐹 ∧ Rel ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)) ∧ 𝜑) → 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
249, 23mpancom 684 1 (𝜑𝐹 = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  wss 3933  cop 4563   class class class wbr 5057  cmpt 5137   × cxp 5546  ccnv 5547  ran crn 5549  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  pi1xfrcnvlem  23587
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