Theorem dftpos2 6494

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 6489	. . 3 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
2	1	reseq2d 5040	. 2 ⊢ (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹 ↾ ◡dom 𝐹))
3		reltpos 6483	. . 3 ⊢ Rel tpos 𝐹
4		resdm 5079	. . 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 6478	. . . 4 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
7	6	reseq1i 5036	. . 3 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹)
8		resco 5269	. . 3 ⊢ ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹))
9		ssun1 3384	. . . . 5 ⊢ ◡dom 𝐹 ⊆ (◡dom 𝐹 ∪ {∅})
10		resmpt 5088	. . . . 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 4917	. . 3 ⊢ (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹)) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))
13	7, 8, 12	3eqtri 2259	. 2 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))
14	2, 5, 13	3eqtr3g 2290	1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})))

Syntax hints:   → wi 4   = wceq 1398   ∪ cun 3211   ⊆ wss 3213  ∅c0 3510  {csn 3691  ∪ cuni 3916   ↦ cmpt 4173  ^◡ccnv 4750  dom cdm 4751   ↾ cres 4753   ∘ ccom 4755  Rel wrel 4756  tpos ctpos 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-tpos 6478

This theorem is referenced by:  tposf12  6502