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

Proof of Theorem intopsn

Step	Hyp	Ref	Expression
1		simpl 475	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ⚬ :(𝐵 × 𝐵)⟶𝐵)
2		id 22	. . . . . 6 ⊢ (𝐵 = {𝑍} → 𝐵 = {𝑍})
3	2	sqxpeqd 5345	. . . . 5 ⊢ (𝐵 = {𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍}))
4	3, 2	feq23d 6252	. . . 4 ⊢ (𝐵 = {𝑍} → ( ⚬ :(𝐵 × 𝐵)⟶𝐵 ↔ ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
5	1, 4	syl5ibcom 237	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} → ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
6		fdm 6265	. . . . . . 7 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → dom ⚬ = (𝐵 × 𝐵))
7	6	eqcomd 2806	. . . . . 6 ⊢ ( ⚬ :(𝐵 × 𝐵)⟶𝐵 → (𝐵 × 𝐵) = dom ⚬ )
8	7	adantr 473	. . . . 5 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 × 𝐵) = dom ⚬ )
9		fdm 6265	. . . . . 6 ⊢ ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → dom ⚬ = ({𝑍} × {𝑍}))
10	9	eqeq2d 2810	. . . . 5 ⊢ ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → ((𝐵 × 𝐵) = dom ⚬ ↔ (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
11	8, 10	syl5ibcom 237	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → (𝐵 × 𝐵) = ({𝑍} × {𝑍})))
12		xpid11 5551	. . . 4 ⊢ ((𝐵 × 𝐵) = ({𝑍} × {𝑍}) ↔ 𝐵 = {𝑍})
13	11, 12	syl6ib 243	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} → 𝐵 = {𝑍}))
14	5, 13	impbid 204	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ :({𝑍} × {𝑍})⟶{𝑍}))
15		simpr 478	. . . 4 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
16		xpsng 6634	. . . 4 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
17	15, 16	sylancom 583	. . 3 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ({𝑍} × {𝑍}) = {⟨𝑍, 𝑍⟩})
18	17	feq2d 6243	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :({𝑍} × {𝑍})⟶{𝑍} ↔ ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍}))
19		opex 5124	. . . 4 ⊢ ⟨𝑍, 𝑍⟩ ∈ V
20		fsng 6632	. . . 4 ⊢ ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍 ∈ 𝐵) → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
21	19, 20	mpan 682	. . 3 ⊢ (𝑍 ∈ 𝐵 → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
22	21	adantl 474	. 2 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → ( ⚬ :{⟨𝑍, 𝑍⟩}⟶{𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
23	14, 18, 22	3bitrd 297	1 ⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))

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