Theorem suppsnop 8119

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

Hypothesis

Ref	Expression
suppsnop.f	⊢ 𝐹 = {⟨𝑋, 𝑌⟩}

Assertion

Ref	Expression
suppsnop	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))

Proof of Theorem suppsnop

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1osng 6810	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
2		f1of 6768	. . . . . . 7 ⊢ ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
3	1, 2	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
4	3	3adant3 1138	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
5		suppsnop.f	. . . . . 6 ⊢ 𝐹 = {⟨𝑋, 𝑌⟩}
6	5	feq1i 6647	. . . . 5 ⊢ (𝐹:{𝑋}⟶{𝑌} ↔ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
7	4, 6	sylibr 235	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝐹:{𝑋}⟶{𝑌})
8		snex 5369	. . . 4 ⊢ {𝑋} ∈ V
9		fex 7171	. . . 4 ⊢ ((𝐹:{𝑋}⟶{𝑌} ∧ {𝑋} ∈ V) → 𝐹 ∈ V)
10	7, 8, 9	sylancl 592	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝐹 ∈ V)
11		simp3 1144	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝑍 ∈ 𝑈)
12		suppval 8103	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
13	10, 11, 12	syl2anc 590	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
14	7	fdmd 6666	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → dom 𝐹 = {𝑋})
15	14	rabeqdv 3406	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
16		sneq 4566	. . . . . 6 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
17	16	imaeq2d 6013	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑋}))
18	17	neeq1d 2993	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹 “ {𝑥}) ≠ {𝑍} ↔ (𝐹 “ {𝑋}) ≠ {𝑍}))
19	18	rabsnif 4656	. . 3 ⊢ {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅)
20	15, 19	eqtrdi 2790	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅))
21	7	ffnd 6657	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝐹 Fn {𝑋})
22		snidg 4593	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
23	22	3ad2ant1 1139	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝑋 ∈ {𝑋})
24		fnsnfv 6907	. . . . . . . 8 ⊢ ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → {(𝐹‘𝑋)} = (𝐹 “ {𝑋}))
25	24	eqcomd 2745	. . . . . . 7 ⊢ ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → (𝐹 “ {𝑋}) = {(𝐹‘𝑋)})
26	21, 23, 25	syl2anc 590	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹 “ {𝑋}) = {(𝐹‘𝑋)})
27	26	neeq1d 2993	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ {(𝐹‘𝑋)} ≠ {𝑍}))
28	5	fveq1i 6829	. . . . . . . 8 ⊢ (𝐹‘𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋)
29		fvsng 7125	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
30	29	3adant3 1138	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
31	28, 30	eqtrid 2786	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹‘𝑋) = 𝑌)
32	31	sneqd 4568	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {(𝐹‘𝑋)} = {𝑌})
33	32	neeq1d 2993	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({(𝐹‘𝑋)} ≠ {𝑍} ↔ {𝑌} ≠ {𝑍}))
34		sneqbg 4775	. . . . . . 7 ⊢ (𝑌 ∈ 𝑊 → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
35	34	3ad2ant2 1140	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
36	35	necon3abid 2970	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({𝑌} ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
37	27, 33, 36	3bitrd 306	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
38	37	ifbid 4479	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(¬ 𝑌 = 𝑍, {𝑋}, ∅))
39		ifnot 4508	. . 3 ⊢ if(¬ 𝑌 = 𝑍, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋})
40	38, 39	eqtrdi 2790	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋}))
41	13, 20, 40	3eqtrd 2778	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  {crab 3391  Vcvv 3431  ∅c0 4262  ifcif 4455  {csn 4556  ⟨cop 4562  dom cdm 5619   “ cima 5622   Fn wfn 6481  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7357   supp csupp 8101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-supp 8102

This theorem is referenced by:  esplyind  33768