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Theorem rntpos 5926
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 2613 . . . . 5 𝑥 ∈ V
21elrn 4625 . . . 4 (𝑥 ∈ ran tpos 𝐹 ↔ ∃𝑦 𝑦tpos 𝐹𝑥)
3 vex 2613 . . . . . . . . 9 𝑦 ∈ V
43, 1breldm 4587 . . . . . . . 8 (𝑦tpos 𝐹𝑥𝑦 ∈ dom tpos 𝐹)
5 dmtpos 5925 . . . . . . . . 9 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
65eleq2d 2152 . . . . . . . 8 (Rel dom 𝐹 → (𝑦 ∈ dom tpos 𝐹𝑦dom 𝐹))
74, 6syl5ib 152 . . . . . . 7 (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥𝑦dom 𝐹))
8 relcnv 4753 . . . . . . . 8 Rel dom 𝐹
9 elrel 4488 . . . . . . . 8 ((Rel dom 𝐹𝑦dom 𝐹) → ∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
108, 9mpan 415 . . . . . . 7 (𝑦dom 𝐹 → ∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
117, 10syl6 33 . . . . . 6 (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥 → ∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩))
12 breq1 3808 . . . . . . . . 9 (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
13 vex 2613 . . . . . . . . . 10 𝑤 ∈ V
14 vex 2613 . . . . . . . . . 10 𝑧 ∈ V
15 brtposg 5923 . . . . . . . . . 10 ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑥 ∈ V) → (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥))
1613, 14, 1, 15mp3an 1269 . . . . . . . . 9 (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥)
1712, 16syl6bb 194 . . . . . . . 8 (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥))
1814, 13opex 4012 . . . . . . . . 9 𝑧, 𝑤⟩ ∈ V
1918, 1brelrn 4615 . . . . . . . 8 (⟨𝑧, 𝑤𝐹𝑥𝑥 ∈ ran 𝐹)
2017, 19syl6bi 161 . . . . . . 7 (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
2120exlimivv 1819 . . . . . 6 (∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
2211, 21syli 37 . . . . 5 (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
2322exlimdv 1742 . . . 4 (Rel dom 𝐹 → (∃𝑦 𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
242, 23syl5bi 150 . . 3 (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹𝑥 ∈ ran 𝐹))
251elrn 4625 . . . 4 (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 𝑦𝐹𝑥)
263, 1breldm 4587 . . . . . . 7 (𝑦𝐹𝑥𝑦 ∈ dom 𝐹)
27 elrel 4488 . . . . . . . 8 ((Rel dom 𝐹𝑦 ∈ dom 𝐹) → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
2827ex 113 . . . . . . 7 (Rel dom 𝐹 → (𝑦 ∈ dom 𝐹 → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩))
2926, 28syl5 32 . . . . . 6 (Rel dom 𝐹 → (𝑦𝐹𝑥 → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩))
30 breq1 3808 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥))
3130, 16syl6bbr 196 . . . . . . . 8 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
3213, 14opex 4012 . . . . . . . . 9 𝑤, 𝑧⟩ ∈ V
3332, 1brelrn 4615 . . . . . . . 8 (⟨𝑤, 𝑧⟩tpos 𝐹𝑥𝑥 ∈ ran tpos 𝐹)
3431, 33syl6bi 161 . . . . . . 7 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3534exlimivv 1819 . . . . . 6 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3629, 35syli 37 . . . . 5 (Rel dom 𝐹 → (𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3736exlimdv 1742 . . . 4 (Rel dom 𝐹 → (∃𝑦 𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3825, 37syl5bi 150 . . 3 (Rel dom 𝐹 → (𝑥 ∈ ran 𝐹𝑥 ∈ ran tpos 𝐹))
3924, 38impbid 127 . 2 (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹𝑥 ∈ ran 𝐹))
4039eqrdv 2081 1 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wex 1422  wcel 1434  Vcvv 2610  cop 3419   class class class wbr 3805  ccnv 4390  dom cdm 4391  ran crn 4392  Rel wrel 4396  tpos ctpos 5913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960  df-tpos 5914
This theorem is referenced by:  tposfo2  5936
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