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Theorem dftpos2 6026
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 6021 . . 3 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
21reseq2d 4713 . 2 (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹dom 𝐹))
3 reltpos 6015 . . 3 Rel tpos 𝐹
4 resdm 4751 . . 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 6010 . . . 4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
76reseq1i 4709 . . 3 (tpos 𝐹dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹)
8 resco 4935 . . 3 ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹))
9 ssun1 3163 . . . . 5 dom 𝐹 ⊆ (dom 𝐹 ∪ {∅})
10 resmpt 4760 . . . . 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 4596 . . 3 (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹)) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
137, 8, 123eqtri 2112 . 2 (tpos 𝐹dom 𝐹) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
142, 5, 133eqtr3g 2143 1 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cun 2997  wss 2999  c0 3286  {csn 3446   cuni 3653  cmpt 3899  ccnv 4437  dom cdm 4438  cres 4440  ccom 4442  Rel wrel 4443  tpos ctpos 6009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023  df-tpos 6010
This theorem is referenced by:  tposf12  6034
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