Theorem dftpos2 7414

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 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))

Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos2

Step	Hyp	Ref	Expression
1		dmtpos 7409	. . 3 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
2	1	reseq2d 5428	. 2 ⊢ (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹 ↾ ◡dom 𝐹))
3		reltpos 7402	. . 3 ⊢ Rel tpos 𝐹
4		resdm 5476	. . 3 ⊢ (Rel tpos 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹)
5	3, 4	ax-mp 5	. 2 ⊢ (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹
6		df-tpos 7397	. . . 4 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
7	6	reseq1i 5424	. . 3 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹)
8		resco 5677	. . 3 ⊢ ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹))
9		ssun1 3809	. . . . 5 ⊢ ◡dom 𝐹 ⊆ (◡dom 𝐹 ∪ {∅})
10		resmpt 5484	. . . . 5 ⊢ (◡dom 𝐹 ⊆ (◡dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹) = (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))
11	9, 10	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹) = (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})
12	11	coeq2i 5315	. . 3 ⊢ (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹)) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))
13	7, 8, 12	3eqtri 2677	. 2 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))
14	2, 5, 13	3eqtr3g 2708	1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))

Syntax hints:   → wi 4   = wceq 1523   ∪ cun 3605   ⊆ wss 3607  ∅c0 3948  {csn 4210  ∪ cuni 4468   ↦ cmpt 4762  ^◡ccnv 5142  dom cdm 5143   ↾ cres 5145   ∘ ccom 5147  Rel wrel 5148  tpos ctpos 7396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-tpos 7397

This theorem is referenced by:  tposf12  7422