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Theorem intopsn 17461
Description: The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 19490. (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 468 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → :(𝐵 × 𝐵)⟶𝐵)
2 id 22 . . . . . 6 (𝐵 = {𝑍} → 𝐵 = {𝑍})
32sqxpeqd 5281 . . . . 5 (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
43, 2feq23d 6180 . . . 4 (𝐵 = {𝑍} → ( :(𝐵 × 𝐵)⟶𝐵 :({𝑍} × {𝑍})⟶{𝑍}))
51, 4syl5ibcom 235 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} → :({𝑍} × {𝑍})⟶{𝑍}))
6 fdm 6191 . . . . . . 7 ( :(𝐵 × 𝐵)⟶𝐵 → dom = (𝐵 × 𝐵))
76eqcomd 2777 . . . . . 6 ( :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom )
87adantr 466 . . . . 5 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 × 𝐵) = dom )
9 fdm 6191 . . . . . 6 ( :({𝑍} × {𝑍})⟶{𝑍} → dom = ({𝑍} × {𝑍}))
109eqeq2d 2781 . . . . 5 ( :({𝑍} × {𝑍})⟶{𝑍} → ((𝐵 × 𝐵) = dom ↔ (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
118, 10syl5ibcom 235 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
12 xpid11 5485 . . . 4 ((𝐵 × 𝐵) = ({𝑍} × {𝑍}) ↔ 𝐵 = {𝑍})
1311, 12syl6ib 241 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} → 𝐵 = {𝑍}))
145, 13impbid 202 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ :({𝑍} × {𝑍})⟶{𝑍}))
15 simpr 471 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → 𝑍𝐵)
16 xpsng 6549 . . . 4 ((𝑍𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1715, 16sylancom 576 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1817feq2d 6171 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} ↔ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19 opex 5060 . . . 4 𝑍, 𝑍⟩ ∈ V
20 fsng 6547 . . . 4 ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍𝐵) → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2119, 20mpan 670 . . 3 (𝑍𝐵 → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2221adantl 467 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2314, 18, 223bitrd 294 1 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4316  cop 4322   × cxp 5247  dom cdm 5249  wf 6027
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 835  df-3an 1073  df-tru 1634  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-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-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-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038
This theorem is referenced by:  mgmb1mgm1  17462
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