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Theorem funopg 6064
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.) (Avoid depending on this detail.)
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 4540 . . . . 5 (𝑢 = 𝐴 → ⟨𝑢, 𝑡⟩ = ⟨𝐴, 𝑡⟩)
21funeqd 6052 . . . 4 (𝑢 = 𝐴 → (Fun ⟨𝑢, 𝑡⟩ ↔ Fun ⟨𝐴, 𝑡⟩))
3 eqeq1 2775 . . . 4 (𝑢 = 𝐴 → (𝑢 = 𝑡𝐴 = 𝑡))
42, 3imbi12d 333 . . 3 (𝑢 = 𝐴 → ((Fun ⟨𝑢, 𝑡⟩ → 𝑢 = 𝑡) ↔ (Fun ⟨𝐴, 𝑡⟩ → 𝐴 = 𝑡)))
5 opeq2 4541 . . . . 5 (𝑡 = 𝐵 → ⟨𝐴, 𝑡⟩ = ⟨𝐴, 𝐵⟩)
65funeqd 6052 . . . 4 (𝑡 = 𝐵 → (Fun ⟨𝐴, 𝑡⟩ ↔ Fun ⟨𝐴, 𝐵⟩))
7 eqeq2 2782 . . . 4 (𝑡 = 𝐵 → (𝐴 = 𝑡𝐴 = 𝐵))
86, 7imbi12d 333 . . 3 (𝑡 = 𝐵 → ((Fun ⟨𝐴, 𝑡⟩ → 𝐴 = 𝑡) ↔ (Fun ⟨𝐴, 𝐵⟩ → 𝐴 = 𝐵)))
9 funrel 6047 . . . . 5 (Fun ⟨𝑢, 𝑡⟩ → Rel ⟨𝑢, 𝑡⟩)
10 vex 3354 . . . . . 6 𝑢 ∈ V
11 vex 3354 . . . . . 6 𝑡 ∈ V
1210, 11relop 5410 . . . . 5 (Rel ⟨𝑢, 𝑡⟩ ↔ ∃𝑥𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
139, 12sylib 208 . . . 4 (Fun ⟨𝑢, 𝑡⟩ → ∃𝑥𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
1410, 11opth 5073 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨{𝑥}, {𝑥, 𝑦}⟩ ↔ (𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}))
15 vex 3354 . . . . . . . . . . . 12 𝑥 ∈ V
1615opid 4560 . . . . . . . . . . 11 𝑥, 𝑥⟩ = {{𝑥}}
1716preq1i 4408 . . . . . . . . . 10 {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}} = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
18 vex 3354 . . . . . . . . . . . 12 𝑦 ∈ V
1915, 18dfop 4539 . . . . . . . . . . 11 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
2019preq2i 4409 . . . . . . . . . 10 {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} = {⟨𝑥, 𝑥⟩, {{𝑥}, {𝑥, 𝑦}}}
21 snex 5037 . . . . . . . . . . 11 {𝑥} ∈ V
22 zfpair2 5036 . . . . . . . . . . 11 {𝑥, 𝑦} ∈ V
2321, 22dfop 4539 . . . . . . . . . 10 ⟨{𝑥}, {𝑥, 𝑦}⟩ = {{{𝑥}}, {{𝑥}, {𝑥, 𝑦}}}
2417, 20, 233eqtr4ri 2804 . . . . . . . . 9 ⟨{𝑥}, {𝑥, 𝑦}⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
2524eqeq2i 2783 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = ⟨{𝑥}, {𝑥, 𝑦}⟩ ↔ ⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩})
2614, 25bitr3i 266 . . . . . . 7 ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) ↔ ⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩})
27 dffun4 6042 . . . . . . . . 9 (Fun ⟨𝑢, 𝑡⟩ ↔ (Rel ⟨𝑢, 𝑡⟩ ∧ ∀𝑧𝑤𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣)))
2827simprbi 484 . . . . . . . 8 (Fun ⟨𝑢, 𝑡⟩ → ∀𝑧𝑤𝑣((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) → 𝑤 = 𝑣))
29 opex 5061 . . . . . . . . . . 11 𝑥, 𝑥⟩ ∈ V
3029prid1 4434 . . . . . . . . . 10 𝑥, 𝑥⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
31 eleq2 2839 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑥⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}))
3230, 31mpbiri 248 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → ⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩)
33 opex 5061 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
3433prid2 4435 . . . . . . . . . 10 𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}
35 eleq2 2839 . . . . . . . . . 10 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩}))
3634, 35mpbiri 248 . . . . . . . . 9 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩)
3732, 36jca 501 . . . . . . . 8 (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩))
38 opeq12 4542 . . . . . . . . . . . . . 14 ((𝑧 = 𝑥𝑤 = 𝑥) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
39383adant3 1126 . . . . . . . . . . . . 13 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑥⟩)
4039eleq1d 2835 . . . . . . . . . . . 12 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → (⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩))
41 opeq12 4542 . . . . . . . . . . . . . 14 ((𝑧 = 𝑥𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
42413adant2 1125 . . . . . . . . . . . . 13 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → ⟨𝑧, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
4342eleq1d 2835 . . . . . . . . . . . 12 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → (⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩ ↔ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩))
4440, 43anbi12d 616 . . . . . . . . . . 11 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → ((⟨𝑧, 𝑤⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑧, 𝑣⟩ ∈ ⟨𝑢, 𝑡⟩) ↔ (⟨𝑥, 𝑥⟩ ∈ ⟨𝑢, 𝑡⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ ⟨𝑢, 𝑡⟩)))
45 eqeq12 2784 . . . . . . . . . . . 12 ((𝑤 = 𝑥𝑣 = 𝑦) → (𝑤 = 𝑣𝑥 = 𝑦))
46453adant1 1124 . . . . . . . . . . 11 ((𝑧 = 𝑥𝑤 = 𝑥𝑣 = 𝑦) → (𝑤 = 𝑣𝑥 = 𝑦))
4744, 46imbi12d 333 . . . . . . . . . 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 3449 . . . . . . . . 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 1572 . . . . . . . 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 40 . . . . . . 7 (Fun ⟨𝑢, 𝑡⟩ → (⟨𝑢, 𝑡⟩ = {⟨𝑥, 𝑥⟩, ⟨𝑥, 𝑦⟩} → 𝑥 = 𝑦))
5126, 50syl5bi 232 . . . . . 6 (Fun ⟨𝑢, 𝑡⟩ → ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑥 = 𝑦))
52 dfsn2 4330 . . . . . . . . . . 11 {𝑥} = {𝑥, 𝑥}
53 preq2 4406 . . . . . . . . . . 11 (𝑥 = 𝑦 → {𝑥, 𝑥} = {𝑥, 𝑦})
5452, 53syl5req 2818 . . . . . . . . . 10 (𝑥 = 𝑦 → {𝑥, 𝑦} = {𝑥})
5554eqeq2d 2781 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑡 = {𝑥, 𝑦} ↔ 𝑡 = {𝑥}))
56 eqtr3 2792 . . . . . . . . . 10 ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥}) → 𝑢 = 𝑡)
5756expcom 398 . . . . . . . . 9 (𝑡 = {𝑥} → (𝑢 = {𝑥} → 𝑢 = 𝑡))
5855, 57syl6bi 243 . . . . . . . 8 (𝑥 = 𝑦 → (𝑡 = {𝑥, 𝑦} → (𝑢 = {𝑥} → 𝑢 = 𝑡)))
5958com13 88 . . . . . . 7 (𝑢 = {𝑥} → (𝑡 = {𝑥, 𝑦} → (𝑥 = 𝑦𝑢 = 𝑡)))
6059imp 393 . . . . . 6 ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → (𝑥 = 𝑦𝑢 = 𝑡))
6151, 60sylcom 30 . . . . 5 (Fun ⟨𝑢, 𝑡⟩ → ((𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑢 = 𝑡))
6261exlimdvv 2014 . . . 4 (Fun ⟨𝑢, 𝑡⟩ → (∃𝑥𝑦(𝑢 = {𝑥} ∧ 𝑡 = {𝑥, 𝑦}) → 𝑢 = 𝑡))
6313, 62mpd 15 . . 3 (Fun ⟨𝑢, 𝑡⟩ → 𝑢 = 𝑡)
644, 8, 63vtocl2g 3421 . 2 ((𝐴𝑉𝐵𝑊) → (Fun ⟨𝐴, 𝐵⟩ → 𝐴 = 𝐵))
65643impia 1109 1 ((𝐴𝑉𝐵𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  {csn 4317  {cpr 4319  cop 4323  Rel wrel 5255  Fun wfun 6024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-fun 6032
This theorem is referenced by: (None)
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