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Theorem rntpos 6163
 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 2693 . . . . 5 𝑥 ∈ V
21elrn 4791 . . . 4 (𝑥 ∈ ran tpos 𝐹 ↔ ∃𝑦 𝑦tpos 𝐹𝑥)
3 vex 2693 . . . . . . . . 9 𝑦 ∈ V
43, 1breldm 4752 . . . . . . . 8 (𝑦tpos 𝐹𝑥𝑦 ∈ dom tpos 𝐹)
5 dmtpos 6162 . . . . . . . . 9 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
65eleq2d 2210 . . . . . . . 8 (Rel dom 𝐹 → (𝑦 ∈ dom tpos 𝐹𝑦dom 𝐹))
74, 6syl5ib 153 . . . . . . 7 (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥𝑦dom 𝐹))
8 relcnv 4926 . . . . . . . 8 Rel dom 𝐹
9 elrel 4650 . . . . . . . 8 ((Rel dom 𝐹𝑦dom 𝐹) → ∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
108, 9mpan 421 . . . . . . 7 (𝑦dom 𝐹 → ∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
117, 10syl6 33 . . . . . 6 (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥 → ∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩))
12 breq1 3941 . . . . . . . . 9 (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
13 vex 2693 . . . . . . . . . 10 𝑤 ∈ V
14 vex 2693 . . . . . . . . . 10 𝑧 ∈ V
15 brtposg 6160 . . . . . . . . . 10 ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑥 ∈ V) → (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥))
1613, 14, 1, 15mp3an 1316 . . . . . . . . 9 (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥)
1712, 16syl6bb 195 . . . . . . . 8 (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥))
1814, 13opex 4160 . . . . . . . . 9 𝑧, 𝑤⟩ ∈ V
1918, 1brelrn 4781 . . . . . . . 8 (⟨𝑧, 𝑤𝐹𝑥𝑥 ∈ ran 𝐹)
2017, 19syl6bi 162 . . . . . . 7 (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
2120exlimivv 1869 . . . . . 6 (∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
2211, 21syli 37 . . . . 5 (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
2322exlimdv 1792 . . . 4 (Rel dom 𝐹 → (∃𝑦 𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
242, 23syl5bi 151 . . 3 (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹𝑥 ∈ ran 𝐹))
251elrn 4791 . . . 4 (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 𝑦𝐹𝑥)
263, 1breldm 4752 . . . . . . 7 (𝑦𝐹𝑥𝑦 ∈ dom 𝐹)
27 elrel 4650 . . . . . . . 8 ((Rel dom 𝐹𝑦 ∈ dom 𝐹) → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
2827ex 114 . . . . . . 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 3941 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥))
3130, 16bitr4di 197 . . . . . . . 8 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
3213, 14opex 4160 . . . . . . . . 9 𝑤, 𝑧⟩ ∈ V
3332, 1brelrn 4781 . . . . . . . 8 (⟨𝑤, 𝑧⟩tpos 𝐹𝑥𝑥 ∈ ran tpos 𝐹)
3431, 33syl6bi 162 . . . . . . 7 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3534exlimivv 1869 . . . . . 6 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3629, 35syli 37 . . . . 5 (Rel dom 𝐹 → (𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3736exlimdv 1792 . . . 4 (Rel dom 𝐹 → (∃𝑦 𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3825, 37syl5bi 151 . . 3 (Rel dom 𝐹 → (𝑥 ∈ ran 𝐹𝑥 ∈ ran tpos 𝐹))
3924, 38impbid 128 . 2 (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹𝑥 ∈ ran 𝐹))
4039eqrdv 2138 1 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2690  ⟨cop 3536   class class class wbr 3938  ◡ccnv 4547  dom cdm 4548  ran crn 4549  Rel wrel 4553  tpos ctpos 6150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-fv 5140  df-tpos 6151 This theorem is referenced by:  tposfo2  6173
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