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Theorem fliftcnv 5356
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 2018 . . . . 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 5353 . . . 4 (φ → ran (x 𝑋 ↦ ⟨B, A⟩) ⊆ (𝑆 × 𝑅))
5 relxp 4370 . . . 4 Rel (𝑆 × 𝑅)
6 relss 4350 . . . 4 (ran (x 𝑋 ↦ ⟨B, A⟩) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (x 𝑋 ↦ ⟨B, A⟩)))
74, 5, 6mpisyl 1311 . . 3 (φ → Rel ran (x 𝑋 ↦ ⟨B, A⟩))
8 relcnv 4626 . . 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 5354 . . . . . 6 (φ → (z𝐹yx 𝑋 (z = A y = B)))
12 vex 2534 . . . . . . 7 y V
13 vex 2534 . . . . . . 7 z V
1412, 13brcnv 4441 . . . . . 6 (y𝐹zz𝐹y)
15 ancom 253 . . . . . . 7 ((y = B z = A) ↔ (z = A y = B))
1615rexbii 2305 . . . . . 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 5354 . . . . 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 3735 . . . 4 (y𝐹z ↔ ⟨y, z 𝐹)
21 df-br 3735 . . . 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 4362 . 2 (((Rel 𝐹 Rel ran (x 𝑋 ↦ ⟨B, A⟩)) φ) → 𝐹 = ran (x 𝑋 ↦ ⟨B, A⟩))
249, 23mpancom 401 1 (φ𝐹 = ran (x 𝑋 ↦ ⟨B, A⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226   wcel 1370  wrex 2281  wss 2890  cop 3349   class class class wbr 3734  cmpt 3788   × cxp 4266  ccnv 4267  ran crn 4269  Rel wrel 4273
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-fv 4833
This theorem is referenced by: (None)
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