MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rntpos Structured version   Visualization version   GIF version

Theorem rntpos 7317
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 3192 . . . . 5 𝑧 ∈ V
21elrn 5331 . . . 4 (𝑧 ∈ ran tpos 𝐹 ↔ ∃𝑤 𝑤tpos 𝐹𝑧)
3 vex 3192 . . . . . . . . 9 𝑤 ∈ V
43, 1breldm 5294 . . . . . . . 8 (𝑤tpos 𝐹𝑧𝑤 ∈ dom tpos 𝐹)
5 dmtpos 7316 . . . . . . . . 9 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
65eleq2d 2684 . . . . . . . 8 (Rel dom 𝐹 → (𝑤 ∈ dom tpos 𝐹𝑤dom 𝐹))
74, 6syl5ib 234 . . . . . . 7 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑤dom 𝐹))
8 relcnv 5467 . . . . . . . 8 Rel dom 𝐹
9 elrel 5188 . . . . . . . 8 ((Rel dom 𝐹𝑤dom 𝐹) → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
108, 9mpan 705 . . . . . . 7 (𝑤dom 𝐹 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
117, 10syl6 35 . . . . . 6 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
12 breq1 4621 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
13 brtpos 7313 . . . . . . . . . 10 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
141, 13ax-mp 5 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
1512, 14syl6bb 276 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
16 opex 4898 . . . . . . . . 9 𝑦, 𝑥⟩ ∈ V
1716, 1brelrn 5321 . . . . . . . 8 (⟨𝑦, 𝑥𝐹𝑧𝑧 ∈ ran 𝐹)
1815, 17syl6bi 243 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
1918exlimivv 1857 . . . . . 6 (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2011, 19syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2120exlimdv 1858 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
222, 21syl5bi 232 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
231elrn 5331 . . . 4 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 𝑤𝐹𝑧)
243, 1breldm 5294 . . . . . . 7 (𝑤𝐹𝑧𝑤 ∈ dom 𝐹)
25 elrel 5188 . . . . . . . 8 ((Rel dom 𝐹𝑤 ∈ dom 𝐹) → ∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩)
2625ex 450 . . . . . . 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 4621 . . . . . . . . 9 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2928, 14syl6bbr 278 . . . . . . . 8 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
30 opex 4898 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
3130, 1brelrn 5321 . . . . . . . 8 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧𝑧 ∈ ran tpos 𝐹)
3229, 31syl6bi 243 . . . . . . 7 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3332exlimivv 1857 . . . . . 6 (∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3427, 33syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3534exlimdv 1858 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3623, 35syl5bi 232 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran 𝐹𝑧 ∈ ran tpos 𝐹))
3722, 36impbid 202 . 2 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
3837eqrdv 2619 1 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wex 1701  wcel 1987  Vcvv 3189  cop 4159   class class class wbr 4618  ccnv 5078  dom cdm 5079  ran crn 5080  Rel wrel 5084  tpos ctpos 7303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-tpos 7304
This theorem is referenced by:  tposfo2  7327  oppchofcl  16832  oyoncl  16842
  Copyright terms: Public domain W3C validator