Theorem intopsn 18837

Description: The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 21039. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)

Assertion

Ref	Expression
intopsn	⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Proof of Theorem intopsn

Step	Hyp	Ref	Expression
1		simpl 488	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ⚬ :(𝐵 × 𝐵)⟶𝐵)
2		id 23	. . . . . 6 ⊢ (𝐵 = {𝑍} → 𝐵 = {𝑍})
3	2	sqxpeqd 5682	. . . . 5 ⊢ (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
4	3, 2	feq23d 6704	. . . 4 ⊢ (𝐵 = {𝑍} → ( ⚬ :(𝐵 × 𝐵)⟶𝐵 ↔ ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
5	1, 4	syl5ibcom 248	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} → ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
6		fdm 6719	. . . . . . 7 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → dom ⚬ = (𝐵 × 𝐵))
7	6	eqcomd 2766	. . . . . 6 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom ⚬ )
8	7	adantr 486	. . . . 5 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 × 𝐵) = dom ⚬ )
9		fdm 6719	. . . . . 6 ⊢ ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → dom ⚬ = ({𝑍} × {𝑍}))
10	9	eqeq2d 2771	. . . . 5 ⊢ ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → ((𝐵 × 𝐵) = dom ⚬ ↔ (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
11	8, 10	syl5ibcom 248	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
12		xpid11 5913	. . . 4 ⊢ ((𝐵 × 𝐵) = ({𝑍} × {𝑍}) ↔ 𝐵 = {𝑍})
13	11, 12	imbitrdi 254	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → 𝐵 = {𝑍}))
14	5, 13	impbid 215	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
15		simpr 490	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
16		xpsng 7140	. . . 4 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
17	15, 16	sylancom 600	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
18	17	feq2d 6693	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} ↔ ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19		opex 5431	. . . 4 ⊢ ⟨𝑍, 𝑍⟩ ∈ V
20		fsng 7138	. . . 4 ⊢ ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍 ∈ 𝐵) → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
21	19, 20	mpan 703	. . 3 ⊢ (𝑍 ∈ 𝐵 → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
22	21	adantl 487	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
23	14, 18, 22	3bitrd 308	1 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {csn 4584  ⟨cop 4834   × cxp 5648  dom cdm 5650  ⟶wf 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545

This theorem is used by:  mgmb1mgm1  18838