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Theorem rntpos 6258
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 2741 . . . . 5 𝑥 ∈ V
21elrn 4871 . . . 4 (𝑥 ∈ ran tpos 𝐹 ↔ ∃𝑦 𝑦tpos 𝐹𝑥)
3 vex 2741 . . . . . . . . 9 𝑦 ∈ V
43, 1breldm 4832 . . . . . . . 8 (𝑦tpos 𝐹𝑥𝑦 ∈ dom tpos 𝐹)
5 dmtpos 6257 . . . . . . . . 9 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
65eleq2d 2247 . . . . . . . 8 (Rel dom 𝐹 → (𝑦 ∈ dom tpos 𝐹𝑦dom 𝐹))
74, 6imbitrid 154 . . . . . . 7 (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥𝑦dom 𝐹))
8 relcnv 5007 . . . . . . . 8 Rel dom 𝐹
9 elrel 4729 . . . . . . . 8 ((Rel dom 𝐹𝑦dom 𝐹) → ∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
108, 9mpan 424 . . . . . . 7 (𝑦dom 𝐹 → ∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
117, 10syl6 33 . . . . . 6 (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥 → ∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩))
12 breq1 4007 . . . . . . . . 9 (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
13 vex 2741 . . . . . . . . . 10 𝑤 ∈ V
14 vex 2741 . . . . . . . . . 10 𝑧 ∈ V
15 brtposg 6255 . . . . . . . . . 10 ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑥 ∈ V) → (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥))
1613, 14, 1, 15mp3an 1337 . . . . . . . . 9 (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥)
1712, 16bitrdi 196 . . . . . . . 8 (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥))
1814, 13opex 4230 . . . . . . . . 9 𝑧, 𝑤⟩ ∈ V
1918, 1brelrn 4861 . . . . . . . 8 (⟨𝑧, 𝑤𝐹𝑥𝑥 ∈ ran 𝐹)
2017, 19syl6bi 163 . . . . . . 7 (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
2120exlimivv 1896 . . . . . 6 (∃𝑤𝑧 𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
2211, 21syli 37 . . . . 5 (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
2322exlimdv 1819 . . . 4 (Rel dom 𝐹 → (∃𝑦 𝑦tpos 𝐹𝑥𝑥 ∈ ran 𝐹))
242, 23biimtrid 152 . . 3 (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹𝑥 ∈ ran 𝐹))
251elrn 4871 . . . 4 (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 𝑦𝐹𝑥)
263, 1breldm 4832 . . . . . . 7 (𝑦𝐹𝑥𝑦 ∈ dom 𝐹)
27 elrel 4729 . . . . . . . 8 ((Rel dom 𝐹𝑦 ∈ dom 𝐹) → ∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
2827ex 115 . . . . . . 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 4007 . . . . . . . . 9 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑧, 𝑤𝐹𝑥))
3130, 16bitr4di 198 . . . . . . . 8 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
3213, 14opex 4230 . . . . . . . . 9 𝑤, 𝑧⟩ ∈ V
3332, 1brelrn 4861 . . . . . . . 8 (⟨𝑤, 𝑧⟩tpos 𝐹𝑥𝑥 ∈ ran tpos 𝐹)
3431, 33syl6bi 163 . . . . . . 7 (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3534exlimivv 1896 . . . . . 6 (∃𝑧𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3629, 35syli 37 . . . . 5 (Rel dom 𝐹 → (𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3736exlimdv 1819 . . . 4 (Rel dom 𝐹 → (∃𝑦 𝑦𝐹𝑥𝑥 ∈ ran tpos 𝐹))
3825, 37biimtrid 152 . . 3 (Rel dom 𝐹 → (𝑥 ∈ ran 𝐹𝑥 ∈ ran tpos 𝐹))
3924, 38impbid 129 . 2 (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹𝑥 ∈ ran 𝐹))
4039eqrdv 2175 1 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2738  cop 3596   class class class wbr 4004  ccnv 4626  dom cdm 4627  ran crn 4628  Rel wrel 4632  tpos ctpos 6245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-tpos 6246
This theorem is referenced by:  tposfo2  6268
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