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Theorem funopg 5157
 Description: A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
funopg ((𝐴𝑉𝐵𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)

Proof of Theorem funopg
Dummy variables 𝑢 𝑡 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3705 . . . . 5 (𝑢 = 𝐴 → ⟨𝑢, 𝑡⟩ = ⟨𝐴, 𝑡⟩)
21funeqd 5145 . . . 4 (𝑢 = 𝐴 → (Fun ⟨𝑢, 𝑡⟩ ↔ Fun ⟨𝐴, 𝑡⟩))
3 eqeq1 2146 . . . 4 (𝑢 = 𝐴 → (𝑢 = 𝑡𝐴 = 𝑡))
42, 3imbi12d 233 . . 3 (𝑢 = 𝐴 → ((Fun ⟨𝑢, 𝑡⟩ → 𝑢 = 𝑡) ↔ (Fun ⟨𝐴, 𝑡⟩ → 𝐴 = 𝑡)))
5 opeq2 3706 . . . . 5 (𝑡 = 𝐵 → ⟨𝐴, 𝑡⟩ = ⟨𝐴, 𝐵⟩)
65funeqd 5145 . . . 4 (𝑡 = 𝐵 → (Fun ⟨𝐴, 𝑡⟩ ↔ Fun ⟨𝐴, 𝐵⟩))
7 eqeq2 2149 . . . 4 (𝑡 = 𝐵 → (𝐴 = 𝑡𝐴 = 𝐵))
86, 7imbi12d 233 . . 3 (𝑡 = 𝐵 → ((Fun ⟨𝐴, 𝑡⟩ → 𝐴 = 𝑡) ↔ (Fun ⟨𝐴, 𝐵⟩ → 𝐴 = 𝐵)))
9 funrel 5140 . . . . 5 (Fun ⟨𝑢, 𝑡⟩ → Rel ⟨𝑢, 𝑡⟩)
10 vex 2689 . . . . . 6 𝑢 ∈ V
11 vex 2689 . . . . . 6 𝑡 ∈ V
1210, 11relop 4689 . . . . 5 (Rel ⟨𝑢, 𝑡⟩ ↔ ∃𝑥𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
139, 12sylib 121 . . . 4 (Fun ⟨𝑢, 𝑡⟩ → ∃𝑥𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
1410, 11opth 4159 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨{𝑥}, {𝑥, 𝑦}⟩ ↔ (𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
15 vex 2689 . . . . . . . . . . . 12 𝑥 ∈ V
1615opid 3723 . . . . . . . . . . 11 𝑥, 𝑥⟩ = {{𝑥}}
1716preq1i 3603 . . . . . . . . . 10 {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}} = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
18 vex 2689 . . . . . . . . . . . 12 𝑦 ∈ V
1915, 18dfop 3704 . . . . . . . . . . 11 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
2019preq2i 3604 . . . . . . . . . 10 {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} = {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}}
2115snex 4109 . . . . . . . . . . 11 {𝑥} ∈ V
22 zfpair2 4132 . . . . . . . . . . 11 {𝑥, 𝑦} ∈ V
2321, 22dfop 3704 . . . . . . . . . 10 ⟨{𝑥}, {𝑥, 𝑦}⟩ = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
2417, 20, 233eqtr4ri 2171 . . . . . . . . 9 ⟨{𝑥}, {𝑥, 𝑦}⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
2524eqeq2i 2150 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨{𝑥}, {𝑥, 𝑦}⟩ ↔ ⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩})
2614, 25bitr3i 185 . . . . . . 7 ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) ↔ ⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩})
27 dffun4 5134 . . . . . . . . 9 (Fun ⟨𝑢, 𝑡⟩ ↔ (Rel ⟨𝑢, 𝑡⟩ ∧ ∀𝑧𝑤𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣)))
2827simprbi 273 . . . . . . . 8 (Fun ⟨𝑢, 𝑡⟩ → ∀𝑧𝑤𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣))
2915, 15opex 4151 . . . . . . . . . . 11 𝑥, 𝑥⟩ ∈ V
3029prid1 3629 . . . . . . . . . 10 𝑥, 𝑥⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
31 eleq2 2203 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑥⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}))
3230, 31mpbiri 167 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → ⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩)
3315, 18opex 4151 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
3433prid2 3630 . . . . . . . . . 10 𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
35 eleq2 2203 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}))
3634, 35mpbiri 167 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩)
3732, 36jca 304 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩))
38 opeq12 3707 . . . . . . . . . . . . . 14 ((𝑧 = 𝑥𝑤 = 𝑥) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
39383adant3 1001 . . . . . . . . . . . . 13 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
4039eleq1d 2208 . . . . . . . . . . . 12 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → (⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩))
41 opeq12 3707 . . . . . . . . . . . . . 14 ((𝑧 = 𝑥𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
42413adant2 1000 . . . . . . . . . . . . 13 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
4342eleq1d 2208 . . . . . . . . . . . 12 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → (⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩))
4440, 43anbi12d 464 . . . . . . . . . . 11 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → ((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) ↔ (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩)))
45 eqeq12 2152 . . . . . . . . . . . 12 ((𝑤 = 𝑥𝑣 = 𝑦) → (𝑤 = 𝑣𝑥 = 𝑦))
46453adant1 999 . . . . . . . . . . 11 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → (𝑤 = 𝑣𝑥 = 𝑦))
4744, 46imbi12d 233 . . . . . . . . . 10 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → (((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣) ↔ ((⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑥 = 𝑦)))
4847spc3gv 2778 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (∀𝑧𝑤𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣) → ((⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑥 = 𝑦)))
4915, 15, 18, 48mp3an 1315 . . . . . . . 8 (∀𝑧𝑤𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣) → ((⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑥 = 𝑦))
5028, 37, 49syl2im 38 . . . . . . 7 (Fun ⟨𝑢, 𝑡⟩ → (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → 𝑥 = 𝑦))
5126, 50syl5bi 151 . . . . . 6 (Fun ⟨𝑢, 𝑡⟩ → ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑥 = 𝑦))
52 dfsn2 3541 . . . . . . . . . . 11 {𝑥} = {𝑥, 𝑥}
53 preq2 3601 . . . . . . . . . . 11 (𝑥 = 𝑦 → {𝑥, 𝑥} = {𝑥, 𝑦})
5452, 53syl5req 2185 . . . . . . . . . 10 (𝑥 = 𝑦 → {𝑥, 𝑦} = {𝑥})
5554eqeq2d 2151 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑡 = {𝑥, 𝑦} ↔ 𝑡 = {𝑥}))
56 eqtr3 2159 . . . . . . . . . 10 ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥}) → 𝑢 = 𝑡)
5756expcom 115 . . . . . . . . 9 (𝑡 = {𝑥} → (𝑢 = {𝑥} → 𝑢 = 𝑡))
5855, 57syl6bi 162 . . . . . . . 8 (𝑥 = 𝑦 → (𝑡 = {𝑥, 𝑦} → (𝑢 = {𝑥} → 𝑢 = 𝑡)))
5958com13 80 . . . . . . 7 (𝑢 = {𝑥} → (𝑡 = {𝑥, 𝑦} → (𝑥 = 𝑦𝑢 = 𝑡)))
6059imp 123 . . . . . 6 ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → (𝑥 = 𝑦𝑢 = 𝑡))
6151, 60sylcom 28 . . . . 5 (Fun ⟨𝑢, 𝑡⟩ → ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑢 = 𝑡))
6261exlimdvv 1869 . . . 4 (Fun ⟨𝑢, 𝑡⟩ → (∃𝑥𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑢 = 𝑡))
6313, 62mpd 13 . . 3 (Fun ⟨𝑢, 𝑡⟩ → 𝑢 = 𝑡)
644, 8, 63vtocl2g 2750 . 2 ((𝐴𝑉𝐵𝑊) → (Fun ⟨𝐴, 𝐵⟩ → 𝐴 = 𝐵))
65643impia 1178 1 ((𝐴𝑉𝐵𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  {csn 3527  {cpr 3528  ⟨cop 3530  Rel wrel 4544  Fun wfun 5117 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-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-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-fun 5125 This theorem is referenced by: (None)
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