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)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Proof of Theorem intopsn

Step	Hyp	Ref	Expression
1		simpl 481	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ⚬ :(𝐵 × 𝐵)⟶𝐵)
2		id 22	. . . . . 6 ⊢ (𝐵 = {𝑍} → 𝐵 = {𝑍})
3	2	sqxpeqd 5710	. . . . 5 ⊢ (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
4	3, 2	feq23d 6718	. . . 4 ⊢ (𝐵 = {𝑍} → ( ⚬ :(𝐵 × 𝐵)⟶𝐵 ↔ ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
5	1, 4	syl5ibcom 244	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} → ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
6		fdm 6732	. . . . . . 7 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → dom ⚬ = (𝐵 × 𝐵))
7	6	eqcomd 2731	. . . . . 6 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom ⚬ )
8	7	adantr 479	. . . . 5 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 × 𝐵) = dom ⚬ )
9		fdm 6732	. . . . . 6 ⊢ ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → dom ⚬ = ({𝑍} × {𝑍}))
10	9	eqeq2d 2736	. . . . 5 ⊢ ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → ((𝐵 × 𝐵) = dom ⚬ ↔ (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
11	8, 10	syl5ibcom 244	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
12		xpid11 5934	. . . 4 ⊢ ((𝐵 × 𝐵) = ({𝑍} × {𝑍}) ↔ 𝐵 = {𝑍})
13	11, 12	imbitrdi 250	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → 𝐵 = {𝑍}))
14	5, 13	impbid 211	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
15		simpr 483	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
16		xpsng 7148	. . . 4 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
17	15, 16	sylancom 586	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
18	17	feq2d 6709	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} ↔ ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19		opex 5466	. . . 4 ⊢ ⟨𝑍, 𝑍⟩ ∈ V
20		fsng 7146	. . . 4 ⊢ ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍 ∈ 𝐵) → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
21	19, 20	mpan 688	. . 3 ⊢ (𝑍 ∈ 𝐵 → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
22	21	adantl 480	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
23	14, 18, 22	3bitrd 304	1 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

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