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Theorem rntpos 7920
 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 3413 . . . . 5 𝑧 ∈ V
21elrn 5738 . . . 4 (𝑧 ∈ ran tpos 𝐹 ↔ ∃𝑤 𝑤tpos 𝐹𝑧)
3 vex 3413 . . . . . . . . 9 𝑤 ∈ V
43, 1breldm 5753 . . . . . . . 8 (𝑤tpos 𝐹𝑧𝑤 ∈ dom tpos 𝐹)
5 dmtpos 7919 . . . . . . . . 9 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
65eleq2d 2837 . . . . . . . 8 (Rel dom 𝐹 → (𝑤 ∈ dom tpos 𝐹𝑤dom 𝐹))
74, 6syl5ib 247 . . . . . . 7 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑤dom 𝐹))
8 relcnv 5943 . . . . . . . 8 Rel dom 𝐹
9 elrel 5644 . . . . . . . 8 ((Rel dom 𝐹𝑤dom 𝐹) → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
108, 9mpan 689 . . . . . . 7 (𝑤dom 𝐹 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
117, 10syl6 35 . . . . . 6 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
12 breq1 5038 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
13 brtpos 7916 . . . . . . . . . 10 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
1413elv 3415 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
1512, 14bitrdi 290 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
16 opex 5327 . . . . . . . . 9 𝑦, 𝑥⟩ ∈ V
1716, 1brelrn 5787 . . . . . . . 8 (⟨𝑦, 𝑥𝐹𝑧𝑧 ∈ ran 𝐹)
1815, 17syl6bi 256 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
1918exlimivv 1933 . . . . . 6 (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2011, 19syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2120exlimdv 1934 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
222, 21syl5bi 245 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
231elrn 5738 . . . 4 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 𝑤𝐹𝑧)
243, 1breldm 5753 . . . . . . 7 (𝑤𝐹𝑧𝑤 ∈ dom 𝐹)
25 elrel 5644 . . . . . . . 8 ((Rel dom 𝐹𝑤 ∈ dom 𝐹) → ∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩)
2625ex 416 . . . . . . 7 (Rel dom 𝐹 → (𝑤 ∈ dom 𝐹 → ∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩))
2724, 26syl5 34 . . . . . 6 (Rel dom 𝐹 → (𝑤𝐹𝑧 → ∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩))
28 breq1 5038 . . . . . . . . 9 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2928, 14bitr4di 292 . . . . . . . 8 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
30 opex 5327 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
3130, 1brelrn 5787 . . . . . . . 8 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧𝑧 ∈ ran tpos 𝐹)
3229, 31syl6bi 256 . . . . . . 7 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3332exlimivv 1933 . . . . . 6 (∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3427, 33syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3534exlimdv 1934 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3623, 35syl5bi 245 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran 𝐹𝑧 ∈ ran tpos 𝐹))
3722, 36impbid 215 . 2 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
3837eqrdv 2756 1 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3409  ⟨cop 4531   class class class wbr 5035  ◡ccnv 5526  dom cdm 5527  ran crn 5528  Rel wrel 5532  tpos ctpos 7906 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-fv 6347  df-tpos 7907 This theorem is referenced by:  tposfo2  7930  oppchofcl  17581  oyoncl  17591
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