Theorem suppsnopdc 6449

Description: The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)

Hypotheses

Ref	Expression
suppsnop.f	⊢ 𝐹 = {⟨𝑋, 𝑌⟩}
suppsnopdc.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
suppsnopdc.y	⊢ (𝜑 → 𝑌 ∈ 𝑊)
suppsnopdc.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)
suppsnopdc.dc	⊢ (𝜑 → DECID 𝑌 = 𝑍)

Assertion

Ref	Expression
suppsnopdc	⊢ (𝜑 → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))

Proof of Theorem suppsnopdc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppsnopdc.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		suppsnopdc.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
3		suppsnopdc.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
4		f1osng 5656	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
5		f1of 5613	. . . . . . . 8 ⊢ ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
6	4, 5	syl 14	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
7	6	3adant3 1044	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
8		suppsnop.f	. . . . . . 7 ⊢ 𝐹 = {⟨𝑋, 𝑌⟩}
9	8	feq1i 5500	. . . . . 6 ⊢ (𝐹:{𝑋}⟶{𝑌} ↔ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
10	7, 9	sylibr 134	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝐹:{𝑋}⟶{𝑌})
11	1, 2, 3, 10	syl3anc 1274	. . . 4 ⊢ (𝜑 → 𝐹:{𝑋}⟶{𝑌})
12		snexg 4296	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → {𝑋} ∈ V)
13	1, 12	syl 14	. . . 4 ⊢ (𝜑 → {𝑋} ∈ V)
14	11, 13	fexd 5915	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
15		suppval 6436	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
16	14, 3, 15	syl2anc 411	. 2 ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
17	10	fdmd 5514	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → dom 𝐹 = {𝑋})
18	17	rabeqdv 2806	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
19		sneq 3699	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
20	19	imaeq2d 5100	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑋}))
21	20	neeq1d 2430	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹 “ {𝑥}) ≠ {𝑍} ↔ (𝐹 “ {𝑋}) ≠ {𝑍}))
22	21	rabsnif 3757	. . . 4 ⊢ {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅)
23	18, 22	eqtrdi 2281	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅))
24	1, 2, 3, 23	syl3anc 1274	. 2 ⊢ (𝜑 → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅))
25	10	ffnd 5508	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝐹 Fn {𝑋})
26		snidg 3717	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
27	26	3ad2ant1 1045	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝑋 ∈ {𝑋})
28		fnsnfv 5735	. . . . . . . . 9 ⊢ ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → {(𝐹‘𝑋)} = (𝐹 “ {𝑋}))
29	28	eqcomd 2238	. . . . . . . 8 ⊢ ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → (𝐹 “ {𝑋}) = {(𝐹‘𝑋)})
30	25, 27, 29	syl2anc 411	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹 “ {𝑋}) = {(𝐹‘𝑋)})
31	30	neeq1d 2430	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ {(𝐹‘𝑋)} ≠ {𝑍}))
32	8	fveq1i 5670	. . . . . . . . 9 ⊢ (𝐹‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋)
33		fvsng 5879	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
34	33	3adant3 1044	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
35	32, 34	eqtrid 2277	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹‘𝑋) = 𝑌)
36	35	sneqd 3701	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {(𝐹‘𝑋)} = {𝑌})
37	36	neeq1d 2430	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({(𝐹‘𝑋)} ≠ {𝑍} ↔ {𝑌} ≠ {𝑍}))
38		sneqbg 3866	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑊 → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
39	38	3ad2ant2 1046	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
40	39	necon3abid 2451	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({𝑌} ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
41	31, 37, 40	3bitrd 214	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
42	41	ifbid 3643	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(¬ 𝑌 = 𝑍, {𝑋}, ∅))
43	1, 2, 3, 42	syl3anc 1274	. . 3 ⊢ (𝜑 → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(¬ 𝑌 = 𝑍, {𝑋}, ∅))
44		suppsnopdc.dc	. . . 4 ⊢ (𝜑 → DECID 𝑌 = 𝑍)
45		ifnotdc 3660	. . . 4 ⊢ (DECID 𝑌 = 𝑍 → if(¬ 𝑌 = 𝑍, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋}))
46	44, 45	syl 14	. . 3 ⊢ (𝜑 → if(¬ 𝑌 = 𝑍, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋}))
47	43, 46	eqtrd 2265	. 2 ⊢ (𝜑 → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋}))
48	16, 24, 47	3eqtrd 2269	1 ⊢ (𝜑 → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 842   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  {crab 2524  Vcvv 2812  ∅c0 3507  ifcif 3619  {csn 3688  ⟨cop 3691  dom cdm 4748   “ cima 4751   Fn wfn 5346  ⟶wf 5347  –1-1-onto→wf1o 5350  ‘cfv 5351  (class class class)co 6049   supp csupp 6434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by:  snopfsuppdc  7251