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Theorem xpcomco 8091
Description: Composition with the bijection of xpcomf1o 8090 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 8090 . . . . . . . . 9 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
3 f1ofun 6177 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → Fun 𝐹)
4 funbrfv2b 6279 . . . . . . . . 9 (Fun 𝐹 → (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤)))
52, 3, 4mp2b 10 . . . . . . . 8 (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤))
6 ancom 465 . . . . . . . 8 ((𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤) ↔ ((𝐹𝑢) = 𝑤𝑢 ∈ dom 𝐹))
7 eqcom 2658 . . . . . . . . 9 ((𝐹𝑢) = 𝑤𝑤 = (𝐹𝑢))
8 f1odm 6179 . . . . . . . . . . 11 (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
92, 8ax-mp 5 . . . . . . . . . 10 dom 𝐹 = (𝐴 × 𝐵)
109eleq2i 2722 . . . . . . . . 9 (𝑢 ∈ dom 𝐹𝑢 ∈ (𝐴 × 𝐵))
117, 10anbi12i 733 . . . . . . . 8 (((𝐹𝑢) = 𝑤𝑢 ∈ dom 𝐹) ↔ (𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)))
125, 6, 113bitri 286 . . . . . . 7 (𝑢𝐹𝑤 ↔ (𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)))
1312anbi1i 731 . . . . . 6 ((𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ((𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)) ∧ 𝑤𝐺𝑣))
14 anass 682 . . . . . 6 (((𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)) ∧ 𝑤𝐺𝑣) ↔ (𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
1513, 14bitri 264 . . . . 5 ((𝑢𝐹𝑤𝑤𝐺𝑣) ↔ (𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
1615exbii 1814 . . . 4 (∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ∃𝑤(𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
17 fvex 6239 . . . . 5 (𝐹𝑢) ∈ V
18 breq1 4688 . . . . . 6 (𝑤 = (𝐹𝑢) → (𝑤𝐺𝑣 ↔ (𝐹𝑢)𝐺𝑣))
1918anbi2d 740 . . . . 5 (𝑤 = (𝐹𝑢) → ((𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣)))
2017, 19ceqsexv 3273 . . . 4 (∃𝑤(𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣))
21 elxp 5165 . . . . . 6 (𝑢 ∈ (𝐴 × 𝐵) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)))
2221anbi1i 731 . . . . 5 ((𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
23 nfcv 2793 . . . . . . 7 𝑧(𝐹𝑢)
24 xpcomco.1 . . . . . . . 8 𝐺 = (𝑦𝐵, 𝑧𝐴𝐶)
25 nfmpt22 6765 . . . . . . . 8 𝑧(𝑦𝐵, 𝑧𝐴𝐶)
2624, 25nfcxfr 2791 . . . . . . 7 𝑧𝐺
27 nfcv 2793 . . . . . . 7 𝑧𝑣
2823, 26, 27nfbr 4732 . . . . . 6 𝑧(𝐹𝑢)𝐺𝑣
292819.41 2141 . . . . 5 (∃𝑧(∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
30 nfcv 2793 . . . . . . . . 9 𝑦(𝐹𝑢)
31 nfmpt21 6764 . . . . . . . . . 10 𝑦(𝑦𝐵, 𝑧𝐴𝐶)
3224, 31nfcxfr 2791 . . . . . . . . 9 𝑦𝐺
33 nfcv 2793 . . . . . . . . 9 𝑦𝑣
3430, 32, 33nfbr 4732 . . . . . . . 8 𝑦(𝐹𝑢)𝐺𝑣
353419.41 2141 . . . . . . 7 (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
36 anass 682 . . . . . . . . 9 (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣)))
37 fveq2 6229 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑧, 𝑦⟩ → (𝐹𝑢) = (𝐹‘⟨𝑧, 𝑦⟩))
38 opelxpi 5182 . . . . . . . . . . . . . . 15 ((𝑧𝐴𝑦𝐵) → ⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵))
39 sneq 4220 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
4039cnveqd 5330 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
4140unieqd 4478 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
42 opswap 5660 . . . . . . . . . . . . . . . . 17 {⟨𝑧, 𝑦⟩} = ⟨𝑦, 𝑧
4341, 42syl6eq 2701 . . . . . . . . . . . . . . . 16 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = ⟨𝑦, 𝑧⟩)
44 opex 4962 . . . . . . . . . . . . . . . 16 𝑦, 𝑧⟩ ∈ V
4543, 1, 44fvmpt 6321 . . . . . . . . . . . . . . 15 (⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
4638, 45syl 17 . . . . . . . . . . . . . 14 ((𝑧𝐴𝑦𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
4737, 46sylan9eq 2705 . . . . . . . . . . . . 13 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → (𝐹𝑢) = ⟨𝑦, 𝑧⟩)
4847breq1d 4695 . . . . . . . . . . . 12 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → ((𝐹𝑢)𝐺𝑣 ↔ ⟨𝑦, 𝑧𝐺𝑣))
49 df-br 4686 . . . . . . . . . . . . . . . 16 (⟨𝑦, 𝑧𝐺𝑣 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺)
50 df-mpt2 6695 . . . . . . . . . . . . . . . . . 18 (𝑦𝐵, 𝑧𝐴𝐶) = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)}
5124, 50eqtri 2673 . . . . . . . . . . . . . . . . 17 𝐺 = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)}
5251eleq2i 2722 . . . . . . . . . . . . . . . 16 (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)})
53 oprabid 6717 . . . . . . . . . . . . . . . 16 (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)} ↔ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶))
5449, 52, 533bitri 286 . . . . . . . . . . . . . . 15 (⟨𝑦, 𝑧𝐺𝑣 ↔ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶))
5554baib 964 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑧𝐴) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
5655ancoms 468 . . . . . . . . . . . . 13 ((𝑧𝐴𝑦𝐵) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
5756adantl 481 . . . . . . . . . . . 12 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
5848, 57bitrd 268 . . . . . . . . . . 11 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → ((𝐹𝑢)𝐺𝑣𝑣 = 𝐶))
5958pm5.32da 674 . . . . . . . . . 10 (𝑢 = ⟨𝑧, 𝑦⟩ → (((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6059pm5.32i 670 . . . . . . . . 9 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣)) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6136, 60bitri 264 . . . . . . . 8 (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6261exbii 1814 . . . . . . 7 (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6335, 62bitr3i 266 . . . . . 6 ((∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6463exbii 1814 . . . . 5 (∃𝑧(∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6522, 29, 643bitr2i 288 . . . 4 ((𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6616, 20, 653bitri 286 . . 3 (∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6766opabbii 4750 . 2 {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
68 df-co 5152 . 2 (𝐺𝐹) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣)}
69 df-mpt2 6695 . . 3 (𝑧𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)}
70 dfoprab2 6743 . . 3 {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
7169, 70eqtri 2673 . 2 (𝑧𝐴, 𝑦𝐵𝐶) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
7267, 68, 713eqtr4i 2683 1 (𝐺𝐹) = (𝑧𝐴, 𝑦𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685  {copab 4745  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  ccom 5147  Fun wfun 5920  1-1-ontowf1o 5925  cfv 5926  {coprab 6691  cmpt2 6692
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-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211
This theorem is referenced by:  omf1o  8104
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