Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tposssxp | Structured version Visualization version GIF version |
Description: The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
tposssxp | ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tpos 7891 | . . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
2 | cossxp 6122 | . . 3 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) | |
3 | 1, 2 | eqsstri 4000 | . 2 ⊢ tpos 𝐹 ⊆ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) |
4 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) | |
5 | 4 | dmmptss 6094 | . . 3 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅}) |
6 | xpss1 5573 | . . 3 ⊢ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅}) → (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹)) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) |
8 | 3, 7 | sstri 3975 | 1 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 {csn 4566 ∪ cuni 4837 ↦ cmpt 5145 × cxp 5552 ◡ccnv 5553 dom cdm 5554 ran crn 5555 ∘ ccom 5558 tpos ctpos 7890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-mpt 5146 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-tpos 7891 |
This theorem is referenced by: reltpos 7896 tposexg 7905 wuntpos 10155 catcoppccl 17367 |
Copyright terms: Public domain | W3C validator |