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Theorem funopsn 6641
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.) (Avoid depending on this detail.)
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 6640 . 2 (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
2 eqeq1 2803 . . . . . . 7 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ ⟨𝑋, 𝑌⟩ = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}))
3 eqcom 2806 . . . . . . 7 (⟨𝑋, 𝑌⟩ = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩)
42, 3syl6bb 279 . . . . . 6 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩))
54adantl 474 . . . . 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 5133 . . . . . . 7 𝑋, 𝑌⟩ ≠ ∅
9 neeq1 3033 . . . . . . . . . . 11 (⟨𝑋, 𝑌⟩ = 𝐹 → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
109eqcoms 2807 . . . . . . . . . 10 (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
11 funrel 6118 . . . . . . . . . . . . . 14 (Fun 𝐹 → Rel 𝐹)
12 reldm0 5546 . . . . . . . . . . . . . 14 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
1311, 12syl 17 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
1413biimprd 240 . . . . . . . . . . . 12 (Fun 𝐹 → (dom 𝐹 = ∅ → 𝐹 = ∅))
1514necon3d 2992 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 ≠ ∅ → dom 𝐹 ≠ ∅))
1615com12 32 . . . . . . . . . 10 (𝐹 ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅))
1710, 16syl6bi 245 . . . . . . . . 9 (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅)))
1817com3l 89 . . . . . . . 8 (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → (𝐹 = ⟨𝑋, 𝑌⟩ → dom 𝐹 ≠ ∅)))
1918impd 399 . . . . . . 7 (⟨𝑋, 𝑌⟩ ≠ ∅ → ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅))
208, 19ax-mp 5 . . . . . 6 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅)
21 fvex 6424 . . . . . . 7 (𝐹𝑥) ∈ V
2221, 6, 7iunopeqop 5177 . . . . . 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 232 . . . 4 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → ∃𝑎dom 𝐹 = {𝑎}))
2524imp 396 . . 3 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → ∃𝑎dom 𝐹 = {𝑎})
26 iuneq1 4724 . . . . . . . . 9 (dom 𝐹 = {𝑎} → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹𝑥)⟩})
27 vex 3388 . . . . . . . . . 10 𝑎 ∈ V
28 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
29 fveq2 6411 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3028, 29opeq12d 4601 . . . . . . . . . . 11 (𝑥 = 𝑎 → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝑎, (𝐹𝑎)⟩)
3130sneqd 4380 . . . . . . . . . 10 (𝑥 = 𝑎 → {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3227, 31iunxsn 4793 . . . . . . . . 9 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩}
3326, 32syl6eq 2849 . . . . . . . 8 (dom 𝐹 = {𝑎} → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3433adantl 474 . . . . . . 7 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3534eqeq2d 2809 . . . . . 6 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}))
36 eqeq1 2803 . . . . . . . . . . 11 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩}))
3736adantl 474 . . . . . . . . . 10 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩}))
38 eqcom 2806 . . . . . . . . . . 11 (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩} ↔ {⟨𝑎, (𝐹𝑎)⟩} = ⟨𝑋, 𝑌⟩)
39 fvex 6424 . . . . . . . . . . . 12 (𝐹𝑎) ∈ V
4027, 39snopeqop 5161 . . . . . . . . . . 11 ({⟨𝑎, (𝐹𝑎)⟩} = ⟨𝑋, 𝑌⟩ ↔ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
4138, 40sylbb 211 . . . . . . . . . 10 (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩} → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
4237, 41syl6bi 245 . . . . . . . . 9 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})))
4342imp 396 . . . . . . . 8 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
44 simpr3 1253 . . . . . . . . . . . 12 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → 𝑋 = {𝑎})
45 simp1 1167 . . . . . . . . . . . . . . . . 17 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → 𝑎 = (𝐹𝑎))
4645eqcomd 2805 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝐹𝑎) = 𝑎)
4746opeq2d 4600 . . . . . . . . . . . . . . 15 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝑎, 𝑎⟩)
4847sneqd 4380 . . . . . . . . . . . . . 14 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → {⟨𝑎, (𝐹𝑎)⟩} = {⟨𝑎, 𝑎⟩})
4948eqeq2d 2809 . . . . . . . . . . . . 13 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
5049biimpac 471 . . . . . . . . . . . 12 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → 𝐹 = {⟨𝑎, 𝑎⟩})
5144, 50jca 508 . . . . . . . . . . 11 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
5251ex 402 . . . . . . . . . 10 (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5352adantl 474 . . . . . . . . 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 444 . . . . . 6 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5735, 56sylbid 232 . . . . 5 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5857impancom 444 . . . 4 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5958eximdv 2013 . . 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 681 1 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  wne 2971  Vcvv 3385  c0 4115  {csn 4368  cop 4374   ciun 4710  dom cdm 5312  Rel wrel 5317  Fun wfun 6095  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109
This theorem is referenced by:  funop  6642  funop1  42138
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