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Theorem dftpos2 5876
Description: Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
dftpos2 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos2
StepHypRef Expression
1 dmtpos 5871 . . 3 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
21reseq2d 4612 . 2 (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹dom 𝐹))
3 reltpos 5865 . . 3 Rel tpos 𝐹
4 resdm 4649 . . 3 (Rel tpos 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹)
53, 4ax-mp 7 . 2 (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹
6 df-tpos 5860 . . . 4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
76reseq1i 4608 . . 3 (tpos 𝐹dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹)
8 resco 4825 . . 3 ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹))
9 ssun1 3106 . . . . 5 dom 𝐹 ⊆ (dom 𝐹 ∪ {∅})
10 resmpt 4656 . . . . 5 (dom 𝐹 ⊆ (dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹) = (𝑥dom 𝐹 {𝑥}))
119, 10ax-mp 7 . . . 4 ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹) = (𝑥dom 𝐹 {𝑥})
1211coeq2i 4496 . . 3 (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹)) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
137, 8, 123eqtri 2064 . 2 (tpos 𝐹dom 𝐹) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
142, 5, 133eqtr3g 2095 1 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  cun 2915  wss 2917  c0 3224  {csn 3375   cuni 3580  cmpt 3818  ccnv 4344  dom cdm 4345  cres 4347  ccom 4349  Rel wrel 4350  tpos ctpos 5859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-tpos 5860
This theorem is referenced by:  tposf12  5884
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