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Theorem fliftcnv 7292
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 2731 . . . . 5 ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩) = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)
2 flift.3 . . . . 5 ((𝜑𝑥𝑋) → 𝐵𝑆)
3 flift.2 . . . . 5 ((𝜑𝑥𝑋) → 𝐴𝑅)
41, 2, 3fliftrel 7289 . . . 4 (𝜑 → ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅))
5 relxp 5687 . . . 4 Rel (𝑆 × 𝑅)
6 relss 5773 . . . 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 6092 . . 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 7290 . . . . . 6 (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑦 = 𝐵)))
12 vex 3477 . . . . . . 7 𝑦 ∈ V
13 vex 3477 . . . . . . 7 𝑧 ∈ V
1412, 13brcnv 5874 . . . . . 6 (𝑦𝐹𝑧𝑧𝐹𝑦)
15 ancom 461 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐴) ↔ (𝑧 = 𝐴𝑦 = 𝐵))
1615rexbii 3093 . . . . . 6 (∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴) ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑦 = 𝐵))
1711, 14, 163bitr4g 313 . . . . 5 (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴)))
181, 2, 3fliftel 7290 . . . . 5 (𝜑 → (𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴)))
1917, 18bitr4d 281 . . . 4 (𝜑 → (𝑦𝐹𝑧𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧))
20 df-br 5142 . . . 4 (𝑦𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐹)
21 df-br 5142 . . . 4 (𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
2219, 20, 213bitr3g 312 . . 3 (𝜑 → (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)))
2322eqrelrdv2 5787 . 2 (((Rel 𝐹 ∧ Rel ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)) ∧ 𝜑) → 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
249, 23mpancom 686 1 (𝜑𝐹 = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3069  wss 3944  cop 4628   class class class wbr 5141  cmpt 5224   × cxp 5667  ccnv 5668  ran crn 5670  Rel wrel 5674
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fun 6534  df-fn 6535  df-f 6536
This theorem is referenced by:  pi1xfrcnvlem  24501
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