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Theorem xpcomco 6713
 Description: Composition with the bijection of xpcomf1o 6712 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 6712 . . . . . . . . 9 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
3 f1ofun 5362 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → Fun 𝐹)
4 funbrfv2b 5459 . . . . . . . . 9 (Fun 𝐹 → (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤)))
52, 3, 4mp2b 8 . . . . . . . 8 (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤))
6 ancom 264 . . . . . . . 8 ((𝑢 ∈ dom 𝐹 ∧ (𝐹𝑢) = 𝑤) ↔ ((𝐹𝑢) = 𝑤𝑢 ∈ dom 𝐹))
7 eqcom 2139 . . . . . . . . 9 ((𝐹𝑢) = 𝑤𝑤 = (𝐹𝑢))
8 f1odm 5364 . . . . . . . . . . 11 (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
92, 8ax-mp 5 . . . . . . . . . 10 dom 𝐹 = (𝐴 × 𝐵)
109eleq2i 2204 . . . . . . . . 9 (𝑢 ∈ dom 𝐹𝑢 ∈ (𝐴 × 𝐵))
117, 10anbi12i 455 . . . . . . . 8 (((𝐹𝑢) = 𝑤𝑢 ∈ dom 𝐹) ↔ (𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)))
125, 6, 113bitri 205 . . . . . . 7 (𝑢𝐹𝑤 ↔ (𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)))
1312anbi1i 453 . . . . . 6 ((𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ((𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)) ∧ 𝑤𝐺𝑣))
14 anass 398 . . . . . 6 (((𝑤 = (𝐹𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)) ∧ 𝑤𝐺𝑣) ↔ (𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
1513, 14bitri 183 . . . . 5 ((𝑢𝐹𝑤𝑤𝐺𝑣) ↔ (𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
1615exbii 1584 . . . 4 (∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ∃𝑤(𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
17 vex 2684 . . . . . . 7 𝑢 ∈ V
181mptfvex 5499 . . . . . . 7 ((∀𝑥 {𝑥} ∈ V ∧ 𝑢 ∈ V) → (𝐹𝑢) ∈ V)
1917, 18mpan2 421 . . . . . 6 (∀𝑥 {𝑥} ∈ V → (𝐹𝑢) ∈ V)
20 vex 2684 . . . . . . . . 9 𝑥 ∈ V
2120snex 4104 . . . . . . . 8 {𝑥} ∈ V
2221cnvex 5072 . . . . . . 7 {𝑥} ∈ V
2322uniex 4354 . . . . . 6 {𝑥} ∈ V
2419, 23mpg 1427 . . . . 5 (𝐹𝑢) ∈ V
25 breq1 3927 . . . . . 6 (𝑤 = (𝐹𝑢) → (𝑤𝐺𝑣 ↔ (𝐹𝑢)𝐺𝑣))
2625anbi2d 459 . . . . 5 (𝑤 = (𝐹𝑢) → ((𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣)))
2724, 26ceqsexv 2720 . . . 4 (∃𝑤(𝑤 = (𝐹𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣))
28 elxp 4551 . . . . . 6 (𝑢 ∈ (𝐴 × 𝐵) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)))
2928anbi1i 453 . . . . 5 ((𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
30 nfcv 2279 . . . . . . 7 𝑧(𝐹𝑢)
31 xpcomco.1 . . . . . . . 8 𝐺 = (𝑦𝐵, 𝑧𝐴𝐶)
32 nfmpo2 5832 . . . . . . . 8 𝑧(𝑦𝐵, 𝑧𝐴𝐶)
3331, 32nfcxfr 2276 . . . . . . 7 𝑧𝐺
34 nfcv 2279 . . . . . . 7 𝑧𝑣
3530, 33, 34nfbr 3969 . . . . . 6 𝑧(𝐹𝑢)𝐺𝑣
363519.41 1664 . . . . 5 (∃𝑧(∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
37 nfcv 2279 . . . . . . . . 9 𝑦(𝐹𝑢)
38 nfmpo1 5831 . . . . . . . . . 10 𝑦(𝑦𝐵, 𝑧𝐴𝐶)
3931, 38nfcxfr 2276 . . . . . . . . 9 𝑦𝐺
40 nfcv 2279 . . . . . . . . 9 𝑦𝑣
4137, 39, 40nfbr 3969 . . . . . . . 8 𝑦(𝐹𝑢)𝐺𝑣
424119.41 1664 . . . . . . 7 (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣))
43 anass 398 . . . . . . . . 9 (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣)))
44 fveq2 5414 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑧, 𝑦⟩ → (𝐹𝑢) = (𝐹‘⟨𝑧, 𝑦⟩))
45 opelxpi 4566 . . . . . . . . . . . . . . 15 ((𝑧𝐴𝑦𝐵) → ⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵))
46 sneq 3533 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
4746cnveqd 4710 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
4847unieqd 3742 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
49 vex 2684 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
50 vex 2684 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
51 opswapg 5020 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ V ∧ 𝑦 ∈ V) → {⟨𝑧, 𝑦⟩} = ⟨𝑦, 𝑧⟩)
5249, 50, 51mp2an 422 . . . . . . . . . . . . . . . . 17 {⟨𝑧, 𝑦⟩} = ⟨𝑦, 𝑧
5348, 52syl6eq 2186 . . . . . . . . . . . . . . . 16 (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = ⟨𝑦, 𝑧⟩)
5450, 49opex 4146 . . . . . . . . . . . . . . . 16 𝑦, 𝑧⟩ ∈ V
5553, 1, 54fvmpt 5491 . . . . . . . . . . . . . . 15 (⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
5645, 55syl 14 . . . . . . . . . . . . . 14 ((𝑧𝐴𝑦𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
5744, 56sylan9eq 2190 . . . . . . . . . . . . 13 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → (𝐹𝑢) = ⟨𝑦, 𝑧⟩)
5857breq1d 3934 . . . . . . . . . . . 12 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → ((𝐹𝑢)𝐺𝑣 ↔ ⟨𝑦, 𝑧𝐺𝑣))
59 df-br 3925 . . . . . . . . . . . . . . . 16 (⟨𝑦, 𝑧𝐺𝑣 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺)
60 df-mpo 5772 . . . . . . . . . . . . . . . . . 18 (𝑦𝐵, 𝑧𝐴𝐶) = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)}
6131, 60eqtri 2158 . . . . . . . . . . . . . . . . 17 𝐺 = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)}
6261eleq2i 2204 . . . . . . . . . . . . . . . 16 (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)})
63 oprabid 5796 . . . . . . . . . . . . . . . 16 (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶)} ↔ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶))
6459, 62, 633bitri 205 . . . . . . . . . . . . . . 15 (⟨𝑦, 𝑧𝐺𝑣 ↔ ((𝑦𝐵𝑧𝐴) ∧ 𝑣 = 𝐶))
6564baib 904 . . . . . . . . . . . . . 14 ((𝑦𝐵𝑧𝐴) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
6665ancoms 266 . . . . . . . . . . . . 13 ((𝑧𝐴𝑦𝐵) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
6766adantl 275 . . . . . . . . . . . 12 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → (⟨𝑦, 𝑧𝐺𝑣𝑣 = 𝐶))
6858, 67bitrd 187 . . . . . . . . . . 11 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) → ((𝐹𝑢)𝐺𝑣𝑣 = 𝐶))
6968pm5.32da 447 . . . . . . . . . 10 (𝑢 = ⟨𝑧, 𝑦⟩ → (((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
7069pm5.32i 449 . . . . . . . . 9 ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ (𝐹𝑢)𝐺𝑣)) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
7143, 70bitri 183 . . . . . . . 8 (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
7271exbii 1584 . . . . . . 7 (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
7342, 72bitr3i 185 . . . . . 6 ((∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
7473exbii 1584 . . . . 5 (∃𝑧(∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧𝐴𝑦𝐵)) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
7529, 36, 743bitr2i 207 . . . 4 ((𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹𝑢)𝐺𝑣) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
7616, 27, 753bitri 205 . . 3 (∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣) ↔ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)))
7776opabbii 3990 . 2 {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
78 df-co 4543 . 2 (𝐺𝐹) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤𝑤𝐺𝑣)}
79 df-mpo 5772 . . 3 (𝑧𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)}
80 dfoprab2 5811 . . 3 {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
8179, 80eqtri 2158 . 2 (𝑧𝐴, 𝑦𝐵𝐶) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧𝐴𝑦𝐵) ∧ 𝑣 = 𝐶))}
8277, 78, 813eqtr4i 2168 1 (𝐺𝐹) = (𝑧𝐴, 𝑦𝐵𝐶)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2681  {csn 3522  ⟨cop 3525  ∪ cuni 3731   class class class wbr 3924  {copab 3983   ↦ cmpt 3984   × cxp 4532  ◡ccnv 4533  dom cdm 4534   ∘ ccom 4538  Fun wfun 5112  –1-1-onto→wf1o 5117  ‘cfv 5118  {coprab 5768   ∈ cmpo 5769 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032 This theorem is referenced by: (None)
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