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Theorem intopsn 18617
Description: The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20676. (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 481 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → :(𝐵 × 𝐵)⟶𝐵)
2 id 22 . . . . . 6 (𝐵 = {𝑍} → 𝐵 = {𝑍})
32sqxpeqd 5710 . . . . 5 (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
43, 2feq23d 6718 . . . 4 (𝐵 = {𝑍} → ( :(𝐵 × 𝐵)⟶𝐵 :({𝑍} × {𝑍})⟶{𝑍}))
51, 4syl5ibcom 244 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} → :({𝑍} × {𝑍})⟶{𝑍}))
6 fdm 6732 . . . . . . 7 ( :(𝐵 × 𝐵)⟶𝐵 → dom = (𝐵 × 𝐵))
76eqcomd 2731 . . . . . 6 ( :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom )
87adantr 479 . . . . 5 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 × 𝐵) = dom )
9 fdm 6732 . . . . . 6 ( :({𝑍} × {𝑍})⟶{𝑍} → dom = ({𝑍} × {𝑍}))
109eqeq2d 2736 . . . . 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 5934 . . . 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 483 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → 𝑍𝐵)
16 xpsng 7148 . . . 4 ((𝑍𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1715, 16sylancom 586 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1817feq2d 6709 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} ↔ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19 opex 5466 . . . 4 𝑍, 𝑍⟩ ∈ V
20 fsng 7146 . . . 4 ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍𝐵) → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2119, 20mpan 688 . . 3 (𝑍𝐵 → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2221adantl 480 . 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 394   = wceq 1533  wcel 2098  Vcvv 3461  {csn 4630  cop 4636   × cxp 5676  dom cdm 5678  wf 6545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556
This theorem is referenced by:  mgmb1mgm1  18618
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