Theorem intopsn 17253

Description: The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 19275. (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 473	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ⚬ :(𝐵 × 𝐵)⟶𝐵)
2		id 22	. . . . . 6 ⊢ (𝐵 = {𝑍} → 𝐵 = {𝑍})
3	2	sqxpeqd 5141	. . . . 5 ⊢ (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
4	3, 2	feq23d 6040	. . . 4 ⊢ (𝐵 = {𝑍} → ( ⚬ :(𝐵 × 𝐵)⟶𝐵 ↔ ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
5	1, 4	syl5ibcom 235	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} → ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
6		fdm 6051	. . . . . . 7 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → dom ⚬ = (𝐵 × 𝐵))
7	6	eqcomd 2628	. . . . . 6 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom ⚬ )
8	7	adantr 481	. . . . 5 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 × 𝐵) = dom ⚬ )
9		fdm 6051	. . . . . 6 ⊢ ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → dom ⚬ = ({𝑍} × {𝑍}))
10	9	eqeq2d 2632	. . . . 5 ⊢ ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → ((𝐵 × 𝐵) = dom ⚬ ↔ (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
11	8, 10	syl5ibcom 235	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
12		xpid11 5347	. . . 4 ⊢ ((𝐵 × 𝐵) = ({𝑍} × {𝑍}) ↔ 𝐵 = {𝑍})
13	11, 12	syl6ib 241	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → 𝐵 = {𝑍}))
14	5, 13	impbid 202	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
15		simpr 477	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
16		xpsng 6406	. . . 4 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
17	15, 16	sylancom 701	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
18	17	feq2d 6031	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} ↔ ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19		opex 4932	. . . 4 ⊢ ⟨𝑍, 𝑍⟩ ∈ V
20		fsng 6404	. . . 4 ⊢ ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍 ∈ 𝐵) → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
21	19, 20	mpan 706	. . 3 ⊢ (𝑍 ∈ 𝐵 → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
22	21	adantl 482	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
23	14, 18, 22	3bitrd 294	1 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ⟨cop 4183   × cxp 5112  dom cdm 5114  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  mgmb1mgm1  17254