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Theorem funopsn 6892
 Description: If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6384, as relsnopg 5653 is to relop 5698. (New usage is discouraged.)
Hypotheses
Ref Expression
funopsn.x 𝑋 ∈ V
funopsn.y 𝑌 ∈ V
Assertion
Ref Expression
funopsn ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
Distinct variable groups:   𝐹,𝑎   𝑋,𝑎   𝑌,𝑎

Proof of Theorem funopsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funiun 6891 . 2 (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
2 eqeq1 2826 . . . . . . 7 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ ⟨𝑋, 𝑌⟩ = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}))
3 eqcom 2829 . . . . . . 7 (⟨𝑋, 𝑌⟩ = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩)
42, 3syl6bb 290 . . . . . 6 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩))
54adantl 485 . . . . 5 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩))
6 funopsn.x . . . . . . . 8 𝑋 ∈ V
7 funopsn.y . . . . . . . 8 𝑌 ∈ V
86, 7opnzi 5343 . . . . . . 7 𝑋, 𝑌⟩ ≠ ∅
9 neeq1 3073 . . . . . . . . . . 11 (⟨𝑋, 𝑌⟩ = 𝐹 → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
109eqcoms 2830 . . . . . . . . . 10 (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
11 funrel 6351 . . . . . . . . . . . . . 14 (Fun 𝐹 → Rel 𝐹)
12 reldm0 5775 . . . . . . . . . . . . . 14 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
1311, 12syl 17 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
1413biimprd 251 . . . . . . . . . . . 12 (Fun 𝐹 → (dom 𝐹 = ∅ → 𝐹 = ∅))
1514necon3d 3032 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 ≠ ∅ → dom 𝐹 ≠ ∅))
1615com12 32 . . . . . . . . . 10 (𝐹 ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅))
1710, 16syl6bi 256 . . . . . . . . 9 (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅)))
1817com3l 89 . . . . . . . 8 (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → (𝐹 = ⟨𝑋, 𝑌⟩ → dom 𝐹 ≠ ∅)))
1918impd 414 . . . . . . 7 (⟨𝑋, 𝑌⟩ ≠ ∅ → ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅))
208, 19ax-mp 5 . . . . . 6 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅)
21 fvex 6665 . . . . . . 7 (𝐹𝑥) ∈ V
2221, 6, 7iunopeqop 5388 . . . . . 6 (dom 𝐹 ≠ ∅ → ( 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩ → ∃𝑎dom 𝐹 = {𝑎}))
2320, 22syl 17 . . . . 5 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ( 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩ → ∃𝑎dom 𝐹 = {𝑎}))
245, 23sylbid 243 . . . 4 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → ∃𝑎dom 𝐹 = {𝑎}))
2524imp 410 . . 3 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → ∃𝑎dom 𝐹 = {𝑎})
26 iuneq1 4910 . . . . . . . . 9 (dom 𝐹 = {𝑎} → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹𝑥)⟩})
27 vex 3472 . . . . . . . . . 10 𝑎 ∈ V
28 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
29 fveq2 6652 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3028, 29opeq12d 4786 . . . . . . . . . . 11 (𝑥 = 𝑎 → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝑎, (𝐹𝑎)⟩)
3130sneqd 4551 . . . . . . . . . 10 (𝑥 = 𝑎 → {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3227, 31iunxsn 4988 . . . . . . . . 9 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩}
3326, 32syl6eq 2873 . . . . . . . 8 (dom 𝐹 = {𝑎} → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3433adantl 485 . . . . . . 7 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3534eqeq2d 2833 . . . . . 6 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}))
36 eqeq1 2826 . . . . . . . . . . 11 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩}))
3736adantl 485 . . . . . . . . . 10 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩}))
38 eqcom 2829 . . . . . . . . . . 11 (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩} ↔ {⟨𝑎, (𝐹𝑎)⟩} = ⟨𝑋, 𝑌⟩)
39 fvex 6665 . . . . . . . . . . . 12 (𝐹𝑎) ∈ V
4027, 39snopeqop 5373 . . . . . . . . . . 11 ({⟨𝑎, (𝐹𝑎)⟩} = ⟨𝑋, 𝑌⟩ ↔ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
4138, 40sylbb 222 . . . . . . . . . 10 (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩} → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
4237, 41syl6bi 256 . . . . . . . . 9 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})))
4342imp 410 . . . . . . . 8 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
44 simpr3 1193 . . . . . . . . . . . 12 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → 𝑋 = {𝑎})
45 simp1 1133 . . . . . . . . . . . . . . . . 17 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → 𝑎 = (𝐹𝑎))
4645eqcomd 2828 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝐹𝑎) = 𝑎)
4746opeq2d 4785 . . . . . . . . . . . . . . 15 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝑎, 𝑎⟩)
4847sneqd 4551 . . . . . . . . . . . . . 14 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → {⟨𝑎, (𝐹𝑎)⟩} = {⟨𝑎, 𝑎⟩})
4948eqeq2d 2833 . . . . . . . . . . . . 13 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
5049biimpac 482 . . . . . . . . . . . 12 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → 𝐹 = {⟨𝑎, 𝑎⟩})
5144, 50jca 515 . . . . . . . . . . 11 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
5251ex 416 . . . . . . . . . 10 (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5352adantl 485 . . . . . . . . 9 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5453a1dd 50 . . . . . . . 8 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))))
5543, 54mpd 15 . . . . . . 7 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5655impancom 455 . . . . . 6 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5735, 56sylbid 243 . . . . 5 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5857impancom 455 . . . 4 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5958eximdv 1918 . . 3 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → (∃𝑎dom 𝐹 = {𝑎} → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
6025, 59mpd 15 . 2 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
611, 60mpidan 688 1 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2114   ≠ wne 3011  Vcvv 3469  ∅c0 4265  {csn 4539  ⟨cop 4545  ∪ ciun 4894  dom cdm 5532  Rel wrel 5537  Fun wfun 6328  ‘cfv 6334 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342 This theorem is referenced by:  funop  6893  funop1  43778
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