Theorem intopsn 18713

Description: The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20861. (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 487	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ⚬ :(𝐵 × 𝐵)⟶𝐵)
2		id 23	. . . . . 6 ⊢ (𝐵 = {𝑍} → 𝐵 = {𝑍})
3	2	sqxpeqd 5695	. . . . 5 ⊢ (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
4	3, 2	feq23d 6702	. . . 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 6717	. . . . . . 7 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → dom ⚬ = (𝐵 × 𝐵))
7	6	eqcomd 2769	. . . . . 6 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom ⚬ )
8	7	adantr 485	. . . . 5 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 × 𝐵) = dom ⚬ )
9		fdm 6717	. . . . . 6 ⊢ ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → dom ⚬ = ({𝑍} × {𝑍}))
10	9	eqeq2d 2774	. . . . 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 5924	. . . 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 489	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
16		xpsng 7137	. . . 4 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
17	15, 16	sylancom 599	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
18	17	feq2d 6691	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} ↔ ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19		opex 5447	. . . 4 ⊢ ⟨𝑍, 𝑍⟩ ∈ V
20		fsng 7135	. . . 4 ⊢ ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍 ∈ 𝐵) → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
21	19, 20	mpan 702	. . 3 ⊢ (𝑍 ∈ 𝐵 → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
22	21	adantl 486	. 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⟩}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4590  ⟨cop 4596   × cxp 5661  dom cdm 5663  ⟶wf 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545

This theorem is referenced by:  mgmb1mgm1  18714