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Mirrors > Home > MPE Home > Th. List > dftpos2 | Structured version Visualization version GIF version |
Description: Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
dftpos2 | ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmtpos 7893 | . . 3 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) | |
2 | 1 | reseq2d 5846 | . 2 ⊢ (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹 ↾ ◡dom 𝐹)) |
3 | reltpos 7886 | . . 3 ⊢ Rel tpos 𝐹 | |
4 | resdm 5890 | . . 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 7881 | . . . 4 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
7 | 6 | reseq1i 5842 | . . 3 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹) |
8 | resco 6096 | . . 3 ⊢ ((𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ↾ ◡dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹)) | |
9 | ssun1 4145 | . . . . 5 ⊢ ◡dom 𝐹 ⊆ (◡dom 𝐹 ∪ {∅}) | |
10 | resmpt 5898 | . . . . 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 5724 | . . 3 ⊢ (𝐹 ∘ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↾ ◡dom 𝐹)) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) |
13 | 7, 8, 12 | 3eqtri 2845 | . 2 ⊢ (tpos 𝐹 ↾ ◡dom 𝐹) = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) |
14 | 2, 5, 13 | 3eqtr3g 2876 | 1 ⊢ (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∪ cun 3931 ⊆ wss 3933 ∅c0 4288 {csn 4557 ∪ cuni 4830 ↦ cmpt 5137 ◡ccnv 5547 dom cdm 5548 ↾ cres 5550 ∘ ccom 5552 Rel wrel 5553 tpos ctpos 7880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-tpos 7881 |
This theorem is referenced by: tposf12 7906 |
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