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Theorem xpcomco 9100
Description: Composition with the bijection of xpcomf1o 9099 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 9099 . . . . . . . . 9 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
3 f1ofun 6850 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → Fun 𝐹)
4 funbrfv2b 6965 . . . . . . . . 9 (Fun 𝐹 → (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤)))
52, 3, 4mp2b 10 . . . . . . . 8 (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤))
6 ancom 460 . . . . . . . 8 ((𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤) ↔ ((𝐹𝑢) = 𝑤𝑢 ∈ dom 𝐹))
7 eqcom 2741 . . . . . . . . 9 ((𝐹𝑢) = 𝑤𝑤 = (𝐹𝑢))
8 f1odm 6852 . . . . . . . . . . 11 (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
92, 8ax-mp 5 . . . . . . . . . 10 dom 𝐹 = (𝐴 × 𝐵)
109eleq2i 2830 . . . . . . . . 9 (𝑢 ∈ dom 𝐹𝑢 ∈ (𝐴 × 𝐵))
117, 10anbi12i 628 . . . . . . . 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 468 . . . . . 6 (((𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)) ∧ 𝑤𝐺𝑣) ↔ (𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
1513, 14bitri 275 . . . . 5 ((𝑢𝐹𝑤𝑤𝐺𝑣) ↔ (𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
1615exbii 1844 . . . 4 (∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ∃𝑤(𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
17 fvex 6919 . . . . 5 (𝐹𝑢) ∈ V
18 breq1 5150 . . . . . 6 (𝑤 = (𝐹𝑢) → (𝑤𝐺𝑣 ↔ (𝐹𝑢)𝐺𝑣))
1918anbi2d 630 . . . . 5 (𝑤 = (𝐹𝑢) → ((𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣)))
2017, 19ceqsexv 3529 . . . 4 (∃𝑤(𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣))
21 elxp 5711 . . . . . 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 2902 . . . . . . 7 𝑧(𝐹𝑢)
24 xpcomco.1 . . . . . . . 8 𝐺 = (𝑦𝐵, 𝑧𝐴𝐶)
25 nfmpo2 7513 . . . . . . . 8 𝑧(𝑦𝐵, 𝑧𝐴𝐶)
2624, 25nfcxfr 2900 . . . . . . 7 𝑧𝐺
27 nfcv 2902 . . . . . . 7 𝑧𝑣
2823, 26, 27nfbr 5194 . . . . . 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 2902 . . . . . . . . 9 𝑦(𝐹𝑢)
31 nfmpo1 7512 . . . . . . . . . 10 𝑦(𝑦𝐵, 𝑧𝐴𝐶)
3224, 31nfcxfr 2900 . . . . . . . . 9 𝑦𝐺
33 nfcv 2902 . . . . . . . . 9 𝑦𝑣
3430, 32, 33nfbr 5194 . . . . . . . 8 𝑦(𝐹𝑢)𝐺𝑣
353419.41 2232 . . . . . . 7 (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
36 anass 468 . . . . . . . . 9 (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣)))
37 fveq2 6906 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑧, 𝑦⟩ → (𝐹𝑢) = (𝐹‘⟨𝑧, 𝑦⟩))
38 opelxpi 5725 . . . . . . . . . . . . . . 15 ((𝑧𝐴𝑦𝐵) → ⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵))
39 sneq 4640 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
4039cnveqd 5888 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
4140unieqd 4924 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
42 opswap 6250 . . . . . . . . . . . . . . . . 17 {⟨𝑧, 𝑦⟩} = ⟨𝑦, 𝑧
4341, 42eqtrdi 2790 . . . . . . . . . . . . . . . 16 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = ⟨𝑦, 𝑧⟩)
44 opex 5474 . . . . . . . . . . . . . . . 16 𝑦, 𝑧⟩ ∈ V
4543, 1, 44fvmpt 7015 . . . . . . . . . . . . . . 15 (⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
4638, 45syl 17 . . . . . . . . . . . . . 14 ((𝑧𝐴𝑦𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
4737, 46sylan9eq 2794 . . . . . . . . . . . . 13 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → (𝐹𝑢) = ⟨𝑦, 𝑧⟩)
4847breq1d 5157 . . . . . . . . . . . 12 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → ((𝐹𝑢)𝐺𝑣 ↔ ⟨𝑦, 𝑧𝐺𝑣))
49 df-br 5148 . . . . . . . . . . . . . . . 16 (⟨𝑦, 𝑧𝐺𝑣 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺)
50 df-mpo 7435 . . . . . . . . . . . . . . . . . 18 (𝑦𝐵, 𝑧𝐴𝐶) = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)}
5124, 50eqtri 2762 . . . . . . . . . . . . . . . . 17 𝐺 = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)}
5251eleq2i 2830 . . . . . . . . . . . . . . . 16 (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)})
53 oprabidw 7461 . . . . . . . . . . . . . . . 16 (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)} ↔ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶))
5449, 52, 533bitri 297 . . . . . . . . . . . . . . 15 (⟨𝑦, 𝑧𝐺𝑣 ↔ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶))
5554baib 535 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑧𝐴) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
5655ancoms 458 . . . . . . . . . . . . 13 ((𝑧𝐴𝑦𝐵) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
5756adantl 481 . . . . . . . . . . . 12 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
5848, 57bitrd 279 . . . . . . . . . . 11 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → ((𝐹𝑢)𝐺𝑣𝑣 = 𝐶))
5958pm5.32da 579 . . . . . . . . . 10 (𝑢 = ⟨𝑧, 𝑦⟩ → (((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6059pm5.32i 574 . . . . . . . . 9 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣)) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6136, 60bitri 275 . . . . . . . 8 (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6261exbii 1844 . . . . . . 7 (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6335, 62bitr3i 277 . . . . . 6 ((∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
6463exbii 1844 . . . . 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 5214 . 2 {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
68 df-co 5697 . 2 (𝐺𝐹) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣)}
69 df-mpo 7435 . . 3 (𝑧𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)}
70 dfoprab2 7490 . . 3 {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
7169, 70eqtri 2762 . 2 (𝑧𝐴, 𝑦𝐵𝐶) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
7267, 68, 713eqtr4i 2772 1 (𝐺𝐹) = (𝑧𝐴, 𝑦𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  {csn 4630  cop 4636   cuni 4911   class class class wbr 5147  {copab 5209  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  ccom 5692  Fun wfun 6556  1-1-ontowf1o 6561  cfv 6562  {coprab 7431  cmpo 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013
This theorem is referenced by:  omf1o  9113
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