MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpcomco Structured version   Visualization version   GIF version

Theorem xpcomco 8829
Description: Composition with the bijection of xpcomf1o 8828 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
xpcomf1o.1 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
xpcomco.1 𝐺 = (𝑦𝐵, 𝑧𝐴𝐶)
Assertion
Ref Expression
xpcomco (𝐺𝐹) = (𝑧𝐴, 𝑦𝐵𝐶)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑦,𝐹,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧)   𝐹(𝑥)   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem xpcomco
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcomf1o.1 . . . . . . . . . 10 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
21xpcomf1o 8828 . . . . . . . . 9 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
3 f1ofun 6715 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → Fun 𝐹)
4 funbrfv2b 6822 . . . . . . . . 9 (Fun 𝐹 → (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤)))
52, 3, 4mp2b 10 . . . . . . . 8 (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤))
6 ancom 461 . . . . . . . 8 ((𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤) ↔ ((𝐹𝑢) = 𝑤𝑢 ∈ dom 𝐹))
7 eqcom 2747 . . . . . . . . 9 ((𝐹𝑢) = 𝑤𝑤 = (𝐹𝑢))
8 f1odm 6717 . . . . . . . . . . 11 (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
92, 8ax-mp 5 . . . . . . . . . 10 dom 𝐹 = (𝐴 × 𝐵)
109eleq2i 2832 . . . . . . . . 9 (𝑢 ∈ dom 𝐹𝑢 ∈ (𝐴 × 𝐵))
117, 10anbi12i 627 . . . . . . . 8 (((𝐹𝑢) = 𝑤𝑢 ∈ dom 𝐹) ↔ (𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)))
125, 6, 113bitri 297 . . . . . . 7 (𝑢𝐹𝑤 ↔ (𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)))
1312anbi1i 624 . . . . . 6 ((𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ((𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)) ∧ 𝑤𝐺𝑣))
14 anass 469 . . . . . 6 (((𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)) ∧ 𝑤𝐺𝑣) ↔ (𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
1513, 14bitri 274 . . . . 5 ((𝑢𝐹𝑤𝑤𝐺𝑣) ↔ (𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
1615exbii 1854 . . . 4 (∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ∃𝑤(𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
17 fvex 6782 . . . . 5 (𝐹𝑢) ∈ V
18 breq1 5082 . . . . . 6 (𝑤 = (𝐹𝑢) → (𝑤𝐺𝑣 ↔ (𝐹𝑢)𝐺𝑣))
1918anbi2d 629 . . . . 5 (𝑤 = (𝐹𝑢) → ((𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣)))
2017, 19ceqsexv 3478 . . . 4 (∃𝑤(𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣))
21 elxp 5612 . . . . . 6 (𝑢 ∈ (𝐴 × 𝐵) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)))
2221anbi1i 624 . . . . 5 ((𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
23 nfcv 2909 . . . . . . 7 𝑧(𝐹𝑢)
24 xpcomco.1 . . . . . . . 8 𝐺 = (𝑦𝐵, 𝑧𝐴𝐶)
25 nfmpo2 7348 . . . . . . . 8 𝑧(𝑦𝐵, 𝑧𝐴𝐶)
2624, 25nfcxfr 2907 . . . . . . 7 𝑧𝐺
27 nfcv 2909 . . . . . . 7 𝑧𝑣
2823, 26, 27nfbr 5126 . . . . . 6 𝑧(𝐹𝑢)𝐺𝑣
292819.41 2232 . . . . 5 (∃𝑧(∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
30 nfcv 2909 . . . . . . . . 9 𝑦(𝐹𝑢)
31 nfmpo1 7347 . . . . . . . . . 10 𝑦(𝑦𝐵, 𝑧𝐴𝐶)
3224, 31nfcxfr 2907 . . . . . . . . 9 𝑦𝐺
33 nfcv 2909 . . . . . . . . 9 𝑦𝑣
3430, 32, 33nfbr 5126 . . . . . . . 8 𝑦(𝐹𝑢)𝐺𝑣
353419.41 2232 . . . . . . 7 (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
36 anass 469 . . . . . . . . 9 (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣)))
37 fveq2 6769 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑧, 𝑦⟩ → (𝐹𝑢) = (𝐹‘⟨𝑧, 𝑦⟩))
38 opelxpi 5626 . . . . . . . . . . . . . . 15 ((𝑧𝐴𝑦𝐵) → ⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵))
39 sneq 4577 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
4039cnveqd 5782 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
4140unieqd 4859 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
42 opswap 6130 . . . . . . . . . . . . . . . . 17 {⟨𝑧, 𝑦⟩} = ⟨𝑦, 𝑧
4341, 42eqtrdi 2796 . . . . . . . . . . . . . . . 16 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = ⟨𝑦, 𝑧⟩)
44 opex 5382 . . . . . . . . . . . . . . . 16 𝑦, 𝑧⟩ ∈ V
4543, 1, 44fvmpt 6870 . . . . . . . . . . . . . . 15 (⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
4638, 45syl 17 . . . . . . . . . . . . . 14 ((𝑧𝐴𝑦𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
4737, 46sylan9eq 2800 . . . . . . . . . . . . 13 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → (𝐹𝑢) = ⟨𝑦, 𝑧⟩)
4847breq1d 5089 . . . . . . . . . . . 12 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → ((𝐹𝑢)𝐺𝑣 ↔ ⟨𝑦, 𝑧𝐺𝑣))
49 df-br 5080 . . . . . . . . . . . . . . . 16 (⟨𝑦, 𝑧𝐺𝑣 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺)
50 df-mpo 7274 . . . . . . . . . . . . . . . . . 18 (𝑦𝐵, 𝑧𝐴𝐶) = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)}
5124, 50eqtri 2768 . . . . . . . . . . . . . . . . 17 𝐺 = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)}
5251eleq2i 2832 . . . . . . . . . . . . . . . 16 (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)})
53 oprabidw 7300 . . . . . . . . . . . . . . . 16 (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)} ↔ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶))
5449, 52, 533bitri 297 . . . . . . . . . . . . . . 15 (⟨𝑦, 𝑧𝐺𝑣 ↔ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶))
5554baib 536 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑧𝐴) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
5655ancoms 459 . . . . . . . . . . . . 13 ((𝑧𝐴𝑦𝐵) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
5756adantl 482 . . . . . . . . . . . 12 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
5848, 57bitrd 278 . . . . . . . . . . 11 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → ((𝐹𝑢)𝐺𝑣𝑣 = 𝐶))
5958pm5.32da 579 . . . . . . . . . 10 (𝑢 = ⟨𝑧, 𝑦⟩ → (((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6059pm5.32i 575 . . . . . . . . 9 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣)) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6136, 60bitri 274 . . . . . . . 8 (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6261exbii 1854 . . . . . . 7 (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6335, 62bitr3i 276 . . . . . 6 ((∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6463exbii 1854 . . . . 5 (∃𝑧(∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6522, 29, 643bitr2i 299 . . . 4 ((𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6616, 20, 653bitri 297 . . 3 (∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6766opabbii 5146 . 2 {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
68 df-co 5598 . 2 (𝐺𝐹) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣)}
69 df-mpo 7274 . . 3 (𝑧𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)}
70 dfoprab2 7325 . . 3 {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
7169, 70eqtri 2768 . 2 (𝑧𝐴, 𝑦𝐵𝐶) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
7267, 68, 713eqtr4i 2778 1 (𝐺𝐹) = (𝑧𝐴, 𝑦𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1542  wex 1786  wcel 2110  {csn 4567  cop 4573   cuni 4845   class class class wbr 5079  {copab 5141  cmpt 5162   × cxp 5587  ccnv 5588  dom cdm 5589  ccom 5593  Fun wfun 6425  1-1-ontowf1o 6430  cfv 6431  {coprab 7270  cmpo 7271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-oprab 7273  df-mpo 7274  df-1st 7822  df-2nd 7823
This theorem is referenced by:  omf1o  8842
  Copyright terms: Public domain W3C validator