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Theorem fliftcnv 6704
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 2771 . . . . 5 ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩) = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)
2 flift.3 . . . . 5 ((𝜑𝑥𝑋) → 𝐵𝑆)
3 flift.2 . . . . 5 ((𝜑𝑥𝑋) → 𝐴𝑅)
41, 2, 3fliftrel 6701 . . . 4 (𝜑 → ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅))
5 relxp 5266 . . . 4 Rel (𝑆 × 𝑅)
6 relss 5346 . . . 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 5644 . . 3 Rel 𝐹
97, 8jctil 509 . 2 (𝜑 → (Rel 𝐹 ∧ Rel ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)))
10 flift.1 . . . . . . 7 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
1110, 3, 2fliftel 6702 . . . . . 6 (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑦 = 𝐵)))
12 vex 3354 . . . . . . 7 𝑦 ∈ V
13 vex 3354 . . . . . . 7 𝑧 ∈ V
1412, 13brcnv 5443 . . . . . 6 (𝑦𝐹𝑧𝑧𝐹𝑦)
15 ancom 452 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐴) ↔ (𝑧 = 𝐴𝑦 = 𝐵))
1615rexbii 3189 . . . . . 6 (∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴) ↔ ∃𝑥𝑋 (𝑧 = 𝐴𝑦 = 𝐵))
1711, 14, 163bitr4g 303 . . . . 5 (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴)))
181, 2, 3fliftel 6702 . . . . 5 (𝜑 → (𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ∃𝑥𝑋 (𝑦 = 𝐵𝑧 = 𝐴)))
1917, 18bitr4d 271 . . . 4 (𝜑 → (𝑦𝐹𝑧𝑦ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧))
20 df-br 4787 . . . 4 (𝑦𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐹)
21 df-br 4787 . . . 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 5359 . 2 (((Rel 𝐹 ∧ Rel ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩)) ∧ 𝜑) → 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
249, 23mpancom 668 1 (𝜑𝐹 = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wrex 3062  wss 3723  cop 4322   class class class wbr 4786  cmpt 4863   × cxp 5247  ccnv 5248  ran crn 5250  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by:  pi1xfrcnvlem  23075
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