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Theorem rntpos 7603
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 3401 . . . . 5 𝑧 ∈ V
21elrn 5574 . . . 4 (𝑧 ∈ ran tpos 𝐹 ↔ ∃𝑤 𝑤tpos 𝐹𝑧)
3 vex 3401 . . . . . . . . 9 𝑤 ∈ V
43, 1breldm 5537 . . . . . . . 8 (𝑤tpos 𝐹𝑧𝑤 ∈ dom tpos 𝐹)
5 dmtpos 7602 . . . . . . . . 9 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
65eleq2d 2878 . . . . . . . 8 (Rel dom 𝐹 → (𝑤 ∈ dom tpos 𝐹𝑤dom 𝐹))
74, 6syl5ib 235 . . . . . . 7 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑤dom 𝐹))
8 relcnv 5720 . . . . . . . 8 Rel dom 𝐹
9 elrel 5431 . . . . . . . 8 ((Rel dom 𝐹𝑤dom 𝐹) → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
108, 9mpan 673 . . . . . . 7 (𝑤dom 𝐹 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
117, 10syl6 35 . . . . . 6 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧 → ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩))
12 breq1 4854 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
13 brtpos 7599 . . . . . . . . . 10 (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
141, 13ax-mp 5 . . . . . . . . 9 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧)
1512, 14syl6bb 278 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
16 opex 5129 . . . . . . . . 9 𝑦, 𝑥⟩ ∈ V
1716, 1brelrn 5564 . . . . . . . 8 (⟨𝑦, 𝑥𝐹𝑧𝑧 ∈ ran 𝐹)
1815, 17syl6bi 244 . . . . . . 7 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
1918exlimivv 2023 . . . . . 6 (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2011, 19syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
2120exlimdv 2024 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤tpos 𝐹𝑧𝑧 ∈ ran 𝐹))
222, 21syl5bi 233 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
231elrn 5574 . . . 4 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 𝑤𝐹𝑧)
243, 1breldm 5537 . . . . . . 7 (𝑤𝐹𝑧𝑤 ∈ dom 𝐹)
25 elrel 5431 . . . . . . . 8 ((Rel dom 𝐹𝑤 ∈ dom 𝐹) → ∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩)
2625ex 399 . . . . . . 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 4854 . . . . . . . . 9 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑦, 𝑥𝐹𝑧))
2928, 14syl6bbr 280 . . . . . . . 8 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧 ↔ ⟨𝑥, 𝑦⟩tpos 𝐹𝑧))
30 opex 5129 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
3130, 1brelrn 5564 . . . . . . . 8 (⟨𝑥, 𝑦⟩tpos 𝐹𝑧𝑧 ∈ ran tpos 𝐹)
3229, 31syl6bi 244 . . . . . . 7 (𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3332exlimivv 2023 . . . . . 6 (∃𝑦𝑥 𝑤 = ⟨𝑦, 𝑥⟩ → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3427, 33syli 39 . . . . 5 (Rel dom 𝐹 → (𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3534exlimdv 2024 . . . 4 (Rel dom 𝐹 → (∃𝑤 𝑤𝐹𝑧𝑧 ∈ ran tpos 𝐹))
3623, 35syl5bi 233 . . 3 (Rel dom 𝐹 → (𝑧 ∈ ran 𝐹𝑧 ∈ ran tpos 𝐹))
3722, 36impbid 203 . 2 (Rel dom 𝐹 → (𝑧 ∈ ran tpos 𝐹𝑧 ∈ ran 𝐹))
3837eqrdv 2811 1 (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  wex 1859  wcel 2157  Vcvv 3398  cop 4383   class class class wbr 4851  ccnv 5317  dom cdm 5318  ran crn 5319  Rel wrel 5323  tpos ctpos 7589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fn 6107  df-fv 6112  df-tpos 7590
This theorem is referenced by:  tposfo2  7613  oppchofcl  17108  oyoncl  17118
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