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Theorem fliftcnv 5378
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 (φ𝐹 = 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.
StepHypRef Expression
1 eqid 2037 . . . . 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)
41, 2, 3fliftrel 5375 . . . 4 (φ → ran (x 𝑋 ↦ ⟨B, A⟩) ⊆ (𝑆 × 𝑅))
5 relxp 4390 . . . 4 Rel (𝑆 × 𝑅)
6 relss 4370 . . . 4 (ran (x 𝑋 ↦ ⟨B, A⟩) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (x 𝑋 ↦ ⟨B, A⟩)))
74, 5, 6mpisyl 1332 . . 3 (φ → Rel ran (x 𝑋 ↦ ⟨B, A⟩))
8 relcnv 4646 . . 3 Rel 𝐹
97, 8jctil 295 . 2 (φ → (Rel 𝐹 Rel ran (x 𝑋 ↦ ⟨B, A⟩)))
10 flift.1 . . . . . . 7 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
1110, 3, 2fliftel 5376 . . . . . 6 (φ → (z𝐹yx 𝑋 (z = A y = B)))
12 vex 2554 . . . . . . 7 y V
13 vex 2554 . . . . . . 7 z V
1412, 13brcnv 4461 . . . . . 6 (y𝐹zz𝐹y)
15 ancom 253 . . . . . . 7 ((y = B z = A) ↔ (z = A y = B))
1615rexbii 2325 . . . . . 6 (x 𝑋 (y = B z = A) ↔ x 𝑋 (z = A y = B))
1711, 14, 163bitr4g 212 . . . . 5 (φ → (y𝐹zx 𝑋 (y = B z = A)))
181, 2, 3fliftel 5376 . . . . 5 (φ → (yran (x 𝑋 ↦ ⟨B, A⟩)zx 𝑋 (y = B z = A)))
1917, 18bitr4d 180 . . . 4 (φ → (y𝐹zyran (x 𝑋 ↦ ⟨B, A⟩)z))
20 df-br 3756 . . . 4 (y𝐹z ↔ ⟨y, z 𝐹)
21 df-br 3756 . . . 4 (yran (x 𝑋 ↦ ⟨B, A⟩)z ↔ ⟨y, z ran (x 𝑋 ↦ ⟨B, A⟩))
2219, 20, 213bitr3g 211 . . 3 (φ → (⟨y, z 𝐹 ↔ ⟨y, z ran (x 𝑋 ↦ ⟨B, A⟩)))
2322eqrelrdv2 4382 . 2 (((Rel 𝐹 Rel ran (x 𝑋 ↦ ⟨B, A⟩)) φ) → 𝐹 = ran (x 𝑋 ↦ ⟨B, A⟩))
249, 23mpancom 399 1 (φ𝐹 = ran (x 𝑋 ↦ ⟨B, A⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wrex 2301  wss 2911  cop 3370   class class class wbr 3755  cmpt 3809   × cxp 4286  ccnv 4287  ran crn 4289  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853
This theorem is referenced by: (None)
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