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Theorem intopsn 18579
Description: The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20540. (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 483 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → :(𝐵 × 𝐵)⟶𝐵)
2 id 22 . . . . . 6 (𝐵 = {𝑍} → 𝐵 = {𝑍})
32sqxpeqd 5708 . . . . 5 (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
43, 2feq23d 6712 . . . 4 (𝐵 = {𝑍} → ( :(𝐵 × 𝐵)⟶𝐵 :({𝑍} × {𝑍})⟶{𝑍}))
51, 4syl5ibcom 244 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} → :({𝑍} × {𝑍})⟶{𝑍}))
6 fdm 6726 . . . . . . 7 ( :(𝐵 × 𝐵)⟶𝐵 → dom = (𝐵 × 𝐵))
76eqcomd 2738 . . . . . 6 ( :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom )
87adantr 481 . . . . 5 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 × 𝐵) = dom )
9 fdm 6726 . . . . . 6 ( :({𝑍} × {𝑍})⟶{𝑍} → dom = ({𝑍} × {𝑍}))
109eqeq2d 2743 . . . . 5 ( :({𝑍} × {𝑍})⟶{𝑍} → ((𝐵 × 𝐵) = dom ↔ (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
118, 10syl5ibcom 244 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
12 xpid11 5931 . . . 4 ((𝐵 × 𝐵) = ({𝑍} × {𝑍}) ↔ 𝐵 = {𝑍})
1311, 12imbitrdi 250 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} → 𝐵 = {𝑍}))
145, 13impbid 211 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ :({𝑍} × {𝑍})⟶{𝑍}))
15 simpr 485 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → 𝑍𝐵)
16 xpsng 7139 . . . 4 ((𝑍𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1715, 16sylancom 588 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1817feq2d 6703 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} ↔ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19 opex 5464 . . . 4 𝑍, 𝑍⟩ ∈ V
20 fsng 7137 . . . 4 ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍𝐵) → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2119, 20mpan 688 . . 3 (𝑍𝐵 → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2221adantl 482 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2314, 18, 223bitrd 304 1 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  {csn 4628  cop 4634   × cxp 5674  dom cdm 5676  wf 6539
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550
This theorem is referenced by:  mgmb1mgm1  18580
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