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Theorem intopsn 17853
Description: The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20034. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)
Assertion
Ref Expression
intopsn (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Proof of Theorem intopsn
StepHypRef Expression
1 simpl 486 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → :(𝐵 × 𝐵)⟶𝐵)
2 id 22 . . . . . 6 (𝐵 = {𝑍} → 𝐵 = {𝑍})
32sqxpeqd 5568 . . . . 5 (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
43, 2feq23d 6490 . . . 4 (𝐵 = {𝑍} → ( :(𝐵 × 𝐵)⟶𝐵 :({𝑍} × {𝑍})⟶{𝑍}))
51, 4syl5ibcom 248 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} → :({𝑍} × {𝑍})⟶{𝑍}))
6 fdm 6503 . . . . . . 7 ( :(𝐵 × 𝐵)⟶𝐵 → dom = (𝐵 × 𝐵))
76eqcomd 2830 . . . . . 6 ( :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom )
87adantr 484 . . . . 5 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 × 𝐵) = dom )
9 fdm 6503 . . . . . 6 ( :({𝑍} × {𝑍})⟶{𝑍} → dom = ({𝑍} × {𝑍}))
109eqeq2d 2835 . . . . 5 ( :({𝑍} × {𝑍})⟶{𝑍} → ((𝐵 × 𝐵) = dom ↔ (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
118, 10syl5ibcom 248 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
12 xpid11 5783 . . . 4 ((𝐵 × 𝐵) = ({𝑍} × {𝑍}) ↔ 𝐵 = {𝑍})
1311, 12syl6ib 254 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} → 𝐵 = {𝑍}))
145, 13impbid 215 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ :({𝑍} × {𝑍})⟶{𝑍}))
15 simpr 488 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → 𝑍𝐵)
16 xpsng 6882 . . . 4 ((𝑍𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1715, 16sylancom 591 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1817feq2d 6481 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} ↔ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19 opex 5337 . . . 4 𝑍, 𝑍⟩ ∈ V
20 fsng 6880 . . . 4 ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍𝐵) → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2119, 20mpan 689 . . 3 (𝑍𝐵 → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2221adantl 485 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2314, 18, 223bitrd 308 1 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3479  {csn 4548  cop 4554   × cxp 5534  dom cdm 5536  wf 6332
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343
This theorem is referenced by:  mgmb1mgm1  17854
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