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Theorem rntpos 5794
 Description: The range of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
rntpos (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Proof of Theorem rntpos
Dummy variables x y w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2538 . . . . 5 x V
21elrn 4504 . . . 4 (x ran tpos 𝐹y ytpos 𝐹x)
3 vex 2538 . . . . . . . . 9 y V
43, 1breldm 4466 . . . . . . . 8 (ytpos 𝐹xy dom tpos 𝐹)
5 dmtpos 5793 . . . . . . . . 9 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
65eleq2d 2089 . . . . . . . 8 (Rel dom 𝐹 → (y dom tpos 𝐹y dom 𝐹))
74, 6syl5ib 143 . . . . . . 7 (Rel dom 𝐹 → (ytpos 𝐹xy dom 𝐹))
8 relcnv 4630 . . . . . . . 8 Rel dom 𝐹
9 elrel 4369 . . . . . . . 8 ((Rel dom 𝐹 y dom 𝐹) → wz y = ⟨w, z⟩)
108, 9mpan 402 . . . . . . 7 (y dom 𝐹wz y = ⟨w, z⟩)
117, 10syl6 29 . . . . . 6 (Rel dom 𝐹 → (ytpos 𝐹xwz y = ⟨w, z⟩))
12 breq1 3741 . . . . . . . . 9 (y = ⟨w, z⟩ → (ytpos 𝐹x ↔ ⟨w, z⟩tpos 𝐹x))
13 vex 2538 . . . . . . . . . 10 w V
14 vex 2538 . . . . . . . . . 10 z V
15 brtposg 5791 . . . . . . . . . 10 ((w V z V x V) → (⟨w, z⟩tpos 𝐹x ↔ ⟨z, w𝐹x))
1613, 14, 1, 15mp3an 1217 . . . . . . . . 9 (⟨w, z⟩tpos 𝐹x ↔ ⟨z, w𝐹x)
1712, 16syl6bb 185 . . . . . . . 8 (y = ⟨w, z⟩ → (ytpos 𝐹x ↔ ⟨z, w𝐹x))
1814, 13opex 3940 . . . . . . . . 9 z, w V
1918, 1brelrn 4494 . . . . . . . 8 (⟨z, w𝐹xx ran 𝐹)
2017, 19syl6bi 152 . . . . . . 7 (y = ⟨w, z⟩ → (ytpos 𝐹xx ran 𝐹))
2120exlimivv 1758 . . . . . 6 (wz y = ⟨w, z⟩ → (ytpos 𝐹xx ran 𝐹))
2211, 21syli 33 . . . . 5 (Rel dom 𝐹 → (ytpos 𝐹xx ran 𝐹))
2322exlimdv 1682 . . . 4 (Rel dom 𝐹 → (y ytpos 𝐹xx ran 𝐹))
242, 23syl5bi 141 . . 3 (Rel dom 𝐹 → (x ran tpos 𝐹x ran 𝐹))
251elrn 4504 . . . 4 (x ran 𝐹y y𝐹x)
263, 1breldm 4466 . . . . . . 7 (y𝐹xy dom 𝐹)
27 elrel 4369 . . . . . . . 8 ((Rel dom 𝐹 y dom 𝐹) → zw y = ⟨z, w⟩)
2827ex 108 . . . . . . 7 (Rel dom 𝐹 → (y dom 𝐹zw y = ⟨z, w⟩))
2926, 28syl5 28 . . . . . 6 (Rel dom 𝐹 → (y𝐹xzw y = ⟨z, w⟩))
30 breq1 3741 . . . . . . . . 9 (y = ⟨z, w⟩ → (y𝐹x ↔ ⟨z, w𝐹x))
3130, 16syl6bbr 187 . . . . . . . 8 (y = ⟨z, w⟩ → (y𝐹x ↔ ⟨w, z⟩tpos 𝐹x))
3213, 14opex 3940 . . . . . . . . 9 w, z V
3332, 1brelrn 4494 . . . . . . . 8 (⟨w, z⟩tpos 𝐹xx ran tpos 𝐹)
3431, 33syl6bi 152 . . . . . . 7 (y = ⟨z, w⟩ → (y𝐹xx ran tpos 𝐹))
3534exlimivv 1758 . . . . . 6 (zw y = ⟨z, w⟩ → (y𝐹xx ran tpos 𝐹))
3629, 35syli 33 . . . . 5 (Rel dom 𝐹 → (y𝐹xx ran tpos 𝐹))
3736exlimdv 1682 . . . 4 (Rel dom 𝐹 → (y y𝐹xx ran tpos 𝐹))
3825, 37syl5bi 141 . . 3 (Rel dom 𝐹 → (x ran 𝐹x ran tpos 𝐹))
3924, 38impbid 120 . 2 (Rel dom 𝐹 → (x ran tpos 𝐹x ran 𝐹))
4039eqrdv 2020 1 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535  ⟨cop 3353   class class class wbr 3738  ◡ccnv 4271  dom cdm 4272  ran crn 4273  Rel wrel 4277  tpos ctpos 5781 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-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-tpos 5782 This theorem is referenced by:  tposfo2  5804
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