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Theorem intopsn 17567
Description: The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 19597. (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 475 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → :(𝐵 × 𝐵)⟶𝐵)
2 id 22 . . . . . 6 (𝐵 = {𝑍} → 𝐵 = {𝑍})
32sqxpeqd 5345 . . . . 5 (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
43, 2feq23d 6252 . . . 4 (𝐵 = {𝑍} → ( :(𝐵 × 𝐵)⟶𝐵 :({𝑍} × {𝑍})⟶{𝑍}))
51, 4syl5ibcom 237 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} → :({𝑍} × {𝑍})⟶{𝑍}))
6 fdm 6265 . . . . . . 7 ( :(𝐵 × 𝐵)⟶𝐵 → dom = (𝐵 × 𝐵))
76eqcomd 2806 . . . . . 6 ( :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom )
87adantr 473 . . . . 5 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 × 𝐵) = dom )
9 fdm 6265 . . . . . 6 ( :({𝑍} × {𝑍})⟶{𝑍} → dom = ({𝑍} × {𝑍}))
109eqeq2d 2810 . . . . 5 ( :({𝑍} × {𝑍})⟶{𝑍} → ((𝐵 × 𝐵) = dom ↔ (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
118, 10syl5ibcom 237 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
12 xpid11 5551 . . . 4 ((𝐵 × 𝐵) = ({𝑍} × {𝑍}) ↔ 𝐵 = {𝑍})
1311, 12syl6ib 243 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} → 𝐵 = {𝑍}))
145, 13impbid 204 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ :({𝑍} × {𝑍})⟶{𝑍}))
15 simpr 478 . . . 4 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → 𝑍𝐵)
16 xpsng 6634 . . . 4 ((𝑍𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1715, 16sylancom 583 . . 3 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
1817feq2d 6243 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :({𝑍} × {𝑍})⟶{𝑍} ↔ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19 opex 5124 . . . 4 𝑍, 𝑍⟩ ∈ V
20 fsng 6632 . . . 4 ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍𝐵) → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2119, 20mpan 682 . . 3 (𝑍𝐵 → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2221adantl 474 . 2 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → ( :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
2314, 18, 223bitrd 297 1 (( :(𝐵 × 𝐵)⟶𝐵𝑍𝐵) → (𝐵 = {𝑍} ↔ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3386  {csn 4369  cop 4375   × cxp 5311  dom cdm 5313  wf 6098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109
This theorem is referenced by:  mgmb1mgm1  17568
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