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Theorem rntpos 7896
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 3503 . . . . 5 𝑧 ∈ V
21elrn 5821 . . . 4 (𝑧 ∈ ran tpos 𝐹 ↔ ∃𝑤 𝑤tpos 𝐹𝑧)
3 vex 3503 . . . . . . . . 9 𝑤 ∈ V
43, 1breldm 5776 . . . . . . . 8 (𝑤tpos 𝐹𝑧𝑤 ∈ dom tpos 𝐹)
5 dmtpos 7895 . . . . . . . . 9 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
65eleq2d 2903 . . . . . . . 8 (Rel dom 𝐹 → (𝑤 ∈ dom tpos 𝐹𝑤dom 𝐹))
74, 6syl5ib 245 . . . . . . 7 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑤dom 𝐹))
8 relcnv 5965 . . . . . . . 8 Rel dom 𝐹
9 elrel 5670 . . . . . . . 8 ((Rel dom 𝐹𝑤dom 𝐹) → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
108, 9mpan 686 . . . . . . 7 (𝑤dom 𝐹 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
117, 10syl6 35 . . . . . 6 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
12 breq1 5066 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
13 brtpos 7892 . . . . . . . . . 10 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
1413elv 3505 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
1512, 14syl6bb 288 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
16 opex 5353 . . . . . . . . 9 𝑦, 𝑥⟩ ∈ V
1716, 1brelrn 5811 . . . . . . . 8 (⟨𝑦, 𝑥𝐹𝑧𝑧 ∈ ran 𝐹)
1815, 17syl6bi 254 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
1918exlimivv 1926 . . . . . 6 (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2011, 19syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2120exlimdv 1927 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
222, 21syl5bi 243 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
231elrn 5821 . . . 4 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 𝑤𝐹𝑧)
243, 1breldm 5776 . . . . . . 7 (𝑤𝐹𝑧𝑤 ∈ dom 𝐹)
25 elrel 5670 . . . . . . . 8 ((Rel dom 𝐹𝑤 ∈ dom 𝐹) → ∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩)
2625ex 413 . . . . . . 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 5066 . . . . . . . . 9 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2928, 14syl6bbr 290 . . . . . . . 8 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
30 opex 5353 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
3130, 1brelrn 5811 . . . . . . . 8 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧𝑧 ∈ ran tpos 𝐹)
3229, 31syl6bi 254 . . . . . . 7 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3332exlimivv 1926 . . . . . 6 (∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3427, 33syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3534exlimdv 1927 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3623, 35syl5bi 243 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran 𝐹𝑧 ∈ ran tpos 𝐹))
3722, 36impbid 213 . 2 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
3837eqrdv 2824 1 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wex 1773  wcel 2107  Vcvv 3500  cop 4570   class class class wbr 5063  ccnv 5553  dom cdm 5554  ran crn 5555  Rel wrel 5559  tpos ctpos 7882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360  df-tpos 7883
This theorem is referenced by:  tposfo2  7906  oppchofcl  17500  oyoncl  17510
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