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Theorem funopsn 6556
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 6555 . 2 (Fun 𝐹𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩})
2 eqeq1 2775 . . . . . . 7 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ ⟨𝑋, 𝑌⟩ = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}))
3 eqcom 2778 . . . . . . 7 (⟨𝑋, 𝑌⟩ = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩)
42, 3syl6bb 276 . . . . . 6 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = ⟨𝑋, 𝑌⟩))
54adantl 467 . . . . 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 5070 . . . . . . 7 𝑋, 𝑌⟩ ≠ ∅
9 neeq1 3005 . . . . . . . . . . 11 (⟨𝑋, 𝑌⟩ = 𝐹 → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
109eqcoms 2779 . . . . . . . . . 10 (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ ↔ 𝐹 ≠ ∅))
11 funrel 6048 . . . . . . . . . . . . . 14 (Fun 𝐹 → Rel 𝐹)
12 reldm0 5481 . . . . . . . . . . . . . 14 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
1311, 12syl 17 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
1413biimprd 238 . . . . . . . . . . . 12 (Fun 𝐹 → (dom 𝐹 = ∅ → 𝐹 = ∅))
1514necon3d 2964 . . . . . . . . . . 11 (Fun 𝐹 → (𝐹 ≠ ∅ → dom 𝐹 ≠ ∅))
1615com12 32 . . . . . . . . . 10 (𝐹 ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅))
1710, 16syl6bi 243 . . . . . . . . 9 (𝐹 = ⟨𝑋, 𝑌⟩ → (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → dom 𝐹 ≠ ∅)))
1817com3l 89 . . . . . . . 8 (⟨𝑋, 𝑌⟩ ≠ ∅ → (Fun 𝐹 → (𝐹 = ⟨𝑋, 𝑌⟩ → dom 𝐹 ≠ ∅)))
1918impd 396 . . . . . . 7 (⟨𝑋, 𝑌⟩ ≠ ∅ → ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅))
208, 19ax-mp 5 . . . . . 6 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → dom 𝐹 ≠ ∅)
21 fvex 6342 . . . . . . 7 (𝐹𝑥) ∈ V
2221, 6, 7iunopeqop 5114 . . . . . 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 230 . . . 4 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → ∃𝑎dom 𝐹 = {𝑎}))
2524imp 393 . . 3 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → ∃𝑎dom 𝐹 = {𝑎})
26 iuneq1 4668 . . . . . . . . 9 (dom 𝐹 = {𝑎} → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹𝑥)⟩})
27 vex 3354 . . . . . . . . . 10 𝑎 ∈ V
28 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
29 fveq2 6332 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3028, 29opeq12d 4547 . . . . . . . . . . 11 (𝑥 = 𝑎 → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝑎, (𝐹𝑎)⟩)
3130sneqd 4328 . . . . . . . . . 10 (𝑥 = 𝑎 → {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3227, 31iunxsn 4737 . . . . . . . . 9 𝑥 ∈ {𝑎} {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩}
3326, 32syl6eq 2821 . . . . . . . 8 (dom 𝐹 = {𝑎} → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3433adantl 467 . . . . . . 7 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} = {⟨𝑎, (𝐹𝑎)⟩})
3534eqeq2d 2781 . . . . . 6 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} ↔ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}))
36 eqeq1 2775 . . . . . . . . . . 11 (𝐹 = ⟨𝑋, 𝑌⟩ → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩}))
3736adantl 467 . . . . . . . . . 10 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ ⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩}))
38 eqcom 2778 . . . . . . . . . . 11 (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩} ↔ {⟨𝑎, (𝐹𝑎)⟩} = ⟨𝑋, 𝑌⟩)
39 fvex 6342 . . . . . . . . . . . 12 (𝐹𝑎) ∈ V
4027, 39snopeqop 5098 . . . . . . . . . . 11 ({⟨𝑎, (𝐹𝑎)⟩} = ⟨𝑋, 𝑌⟩ ↔ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
4138, 40sylbb 209 . . . . . . . . . 10 (⟨𝑋, 𝑌⟩ = {⟨𝑎, (𝐹𝑎)⟩} → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
4237, 41syl6bi 243 . . . . . . . . 9 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})))
4342imp 393 . . . . . . . 8 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = {⟨𝑎, (𝐹𝑎)⟩}) → (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}))
44 simpr3 1237 . . . . . . . . . . . 12 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → 𝑋 = {𝑎})
45 simp1 1130 . . . . . . . . . . . . . . . . 17 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → 𝑎 = (𝐹𝑎))
4645eqcomd 2777 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝐹𝑎) = 𝑎)
4746opeq2d 4546 . . . . . . . . . . . . . . 15 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → ⟨𝑎, (𝐹𝑎)⟩ = ⟨𝑎, 𝑎⟩)
4847sneqd 4328 . . . . . . . . . . . . . 14 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → {⟨𝑎, (𝐹𝑎)⟩} = {⟨𝑎, 𝑎⟩})
4948eqeq2d 2781 . . . . . . . . . . . . 13 ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ↔ 𝐹 = {⟨𝑎, 𝑎⟩}))
5049biimpac 464 . . . . . . . . . . . 12 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → 𝐹 = {⟨𝑎, 𝑎⟩})
5144, 50jca 495 . . . . . . . . . . 11 ((𝐹 = {⟨𝑎, (𝐹𝑎)⟩} ∧ (𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎})) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
5251ex 397 . . . . . . . . . 10 (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → ((𝑎 = (𝐹𝑎) ∧ 𝑋 = 𝑌𝑋 = {𝑎}) → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5352adantl 467 . . . . . . . . 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 439 . . . . . 6 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = {⟨𝑎, (𝐹𝑎)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5735, 56sylbid 230 . . . . 5 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ dom 𝐹 = {𝑎}) → (𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5857impancom 439 . . . 4 (((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) ∧ 𝐹 = 𝑥 ∈ dom 𝐹{⟨𝑥, (𝐹𝑥)⟩}) → (dom 𝐹 = {𝑎} → (𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩})))
5958eximdv 1998 . . 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 661 1 ((Fun 𝐹𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wne 2943  Vcvv 3351  c0 4063  {csn 4316  cop 4322   ciun 4654  dom cdm 5249  Rel wrel 5254  Fun wfun 6025  cfv 6031
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 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-fal 1637  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-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  funop  6557  funop1  41825
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