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Theorem funopsn 6453
Description: If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
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 6452 . . 3 (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
2 eqeq1 2655 . . . . . . . . . 10 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ ⟨𝑋, 𝑌⟩ = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}))
3 eqcom 2658 . . . . . . . . . 10 (⟨𝑋, 𝑌⟩ = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩)
42, 3syl6bb 276 . . . . . . . . 9 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩))
54adantl 481 . . . . . . . 8 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩))
6 funopsn.x . . . . . . . . . . 11 𝑋 ∈ V
7 funopsn.y . . . . . . . . . . 11 𝑌 ∈ V
86, 7opnzi 4972 . . . . . . . . . 10 𝑋, 𝑌⟩ ≠ ∅
9 neeq1 2885 . . . . . . . . . . . . . 14 (⟨𝑋, 𝑌⟩ = 𝐹 → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
109eqcoms 2659 . . . . . . . . . . . . 13 (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
11 funrel 5943 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 → Rel 𝐹)
12 reldm0 5375 . . . . . . . . . . . . . . . . 17 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
1311, 12syl 17 . . . . . . . . . . . . . . . 16 (Fun 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
1413biimprd 238 . . . . . . . . . . . . . . 15 (Fun 𝐹 → (dom 𝐹 = ∅ → 𝐹 = ∅))
1514necon3d 2844 . . . . . . . . . . . . . 14 (Fun 𝐹 → (𝐹 ≠ ∅ → dom 𝐹 ≠ ∅))
1615com12 32 . . . . . . . . . . . . 13 (𝐹 ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅))
1710, 16syl6bi 243 . . . . . . . . . . . 12 (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅)))
1817com3l 89 . . . . . . . . . . 11 (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → (𝐹 = ⟨𝑋, 𝑌⟩ → dom 𝐹 ≠ ∅)))
1918impd 446 . . . . . . . . . 10 (⟨𝑋, 𝑌⟩ ≠ ∅ → ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅))
208, 19ax-mp 5 . . . . . . . . 9 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅)
21 fvex 6239 . . . . . . . . . 10 (𝐹𝑥) ∈ V
2221, 6, 7iunopeqop 5010 . . . . . . . . 9 (dom 𝐹 ≠ ∅ → ( 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩ → ∃𝑎dom 𝐹 = {𝑎}))
2320, 22syl 17 . . . . . . . 8 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ( 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩ → ∃𝑎dom 𝐹 = {𝑎}))
245, 23sylbid 230 . . . . . . 7 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → ∃𝑎dom 𝐹 = {𝑎}))
2524imp 444 . . . . . 6 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → ∃𝑎dom 𝐹 = {𝑎})
26 iuneq1 4566 . . . . . . . . . . . 12 (dom 𝐹 = {𝑎} → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹𝑥)⟩})
27 vex 3234 . . . . . . . . . . . . 13 𝑎 ∈ V
28 id 22 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎𝑥 = 𝑎)
29 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3028, 29opeq12d 4441 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝑎, (𝐹𝑎)⟩)
3130sneqd 4222 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3227, 31iunxsn 4635 . . . . . . . . . . . 12 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩}
3326, 32syl6eq 2701 . . . . . . . . . . 11 (dom 𝐹 = {𝑎} → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3433adantl 481 . . . . . . . . . 10 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3534eqeq2d 2661 . . . . . . . . 9 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}))
36 eqeq1 2655 . . . . . . . . . . . . . 14 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩}))
3736adantl 481 . . . . . . . . . . . . 13 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩}))
38 eqcom 2658 . . . . . . . . . . . . . 14 (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩} ↔ {⟨𝑎, (𝐹𝑎)⟩} = ⟨𝑋, 𝑌⟩)
39 fvex 6239 . . . . . . . . . . . . . . 15 (𝐹𝑎) ∈ V
4027, 39, 6, 7snopeqop 4998 . . . . . . . . . . . . . 14 ({⟨𝑎, (𝐹𝑎)⟩} = ⟨𝑋, 𝑌⟩ ↔ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
4138, 40sylbb 209 . . . . . . . . . . . . 13 (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩} → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
4237, 41syl6bi 243 . . . . . . . . . . . 12 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})))
4342imp 444 . . . . . . . . . . 11 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
44 simpr3 1089 . . . . . . . . . . . . . . 15 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → 𝑋 = {𝑎})
45 simp1 1081 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → 𝑎 = (𝐹𝑎))
4645eqcomd 2657 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝐹𝑎) = 𝑎)
4746opeq2d 4440 . . . . . . . . . . . . . . . . . 18 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝑎, 𝑎⟩)
4847sneqd 4222 . . . . . . . . . . . . . . . . 17 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → {⟨𝑎, (𝐹𝑎)⟩} = {⟨𝑎, 𝑎⟩})
4948eqeq2d 2661 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
5049biimpac 502 . . . . . . . . . . . . . . 15 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → 𝐹 = {⟨𝑎, 𝑎⟩})
5144, 50jca 553 . . . . . . . . . . . . . 14 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
5251ex 449 . . . . . . . . . . . . 13 (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5352adantl 481 . . . . . . . . . . . 12 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5453a1dd 50 . . . . . . . . . . 11 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))))
5543, 54mpd 15 . . . . . . . . . 10 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5655impancom 455 . . . . . . . . 9 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5735, 56sylbid 230 . . . . . . . 8 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5857impancom 455 . . . . . . 7 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5958eximdv 1886 . . . . . 6 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → (∃𝑎dom 𝐹 = {𝑎} → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
6025, 59mpd 15 . . . . 5 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
6160expcom 450 . . . 4 (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
6261expd 451 . . 3 (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → (Fun 𝐹 → (𝐹 = ⟨𝑋, 𝑌⟩ → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))))
631, 62mpcom 38 . 2 (Fun 𝐹 → (𝐹 = ⟨𝑋, 𝑌⟩ → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
6463imp 444 1 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  Vcvv 3231  c0 3948  {csn 4210  cop 4216   ciun 4552  dom cdm 5143  Rel wrel 5148  Fun wfun 5920  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  funop  6454  funop1  41625
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