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Theorem suppsnop 7354
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 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))

Proof of Theorem suppsnop
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 f1osng 6215 . . . . . . 7 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
2 f1of 6175 . . . . . . 7 ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
31, 2syl 17 . . . . . 6 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
433adant3 1101 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
5 suppsnop.f . . . . . 6 𝐹 = {⟨𝑋, 𝑌⟩}
65feq1i 6074 . . . . 5 (𝐹:{𝑋}⟶{𝑌} ↔ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
74, 6sylibr 224 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹:{𝑋}⟶{𝑌})
8 snex 4938 . . . . 5 {𝑋} ∈ V
98a1i 11 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {𝑋} ∈ V)
10 fex 6530 . . . 4 ((𝐹:{𝑋}⟶{𝑌} ∧ {𝑋} ∈ V) → 𝐹 ∈ V)
117, 9, 10syl2anc 694 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 ∈ V)
12 simp3 1083 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝑍𝑈)
13 suppval 7342 . . 3 ((𝐹 ∈ V ∧ 𝑍𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
1411, 12, 13syl2anc 694 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
155a1i 11 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 = {⟨𝑋, 𝑌⟩})
1615dmeqd 5358 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom 𝐹 = dom {⟨𝑋, 𝑌⟩})
17 dmsnopg 5642 . . . . . 6 (𝑌𝑊 → dom {⟨𝑋, 𝑌⟩} = {𝑋})
18173ad2ant2 1103 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom {⟨𝑋, 𝑌⟩} = {𝑋})
1916, 18eqtrd 2685 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom 𝐹 = {𝑋})
20 rabeq 3223 . . . 4 (dom 𝐹 = {𝑋} → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
2119, 20syl 17 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
22 sneq 4220 . . . . . 6 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2322imaeq2d 5501 . . . . 5 (𝑥 = 𝑋 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑋}))
2423neeq1d 2882 . . . 4 (𝑥 = 𝑋 → ((𝐹 “ {𝑥}) ≠ {𝑍} ↔ (𝐹 “ {𝑋}) ≠ {𝑍}))
2524rabsnif 4290 . . 3 {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅)
2621, 25syl6eq 2701 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅))
27 fnsng 5976 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
28273adant3 1101 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
295eqcomi 2660 . . . . . . . . . 10 {⟨𝑋, 𝑌⟩} = 𝐹
3029a1i 11 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} = 𝐹)
3130fneq1d 6019 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({⟨𝑋, 𝑌⟩} Fn {𝑋} ↔ 𝐹 Fn {𝑋}))
3228, 31mpbid 222 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 Fn {𝑋})
33 snidg 4239 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
34333ad2ant1 1102 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝑋 ∈ {𝑋})
35 fnsnfv 6297 . . . . . . . 8 ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → {(𝐹𝑋)} = (𝐹 “ {𝑋}))
3635eqcomd 2657 . . . . . . 7 ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → (𝐹 “ {𝑋}) = {(𝐹𝑋)})
3732, 34, 36syl2anc 694 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 “ {𝑋}) = {(𝐹𝑋)})
3837neeq1d 2882 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ {(𝐹𝑋)} ≠ {𝑍}))
395fveq1i 6230 . . . . . . . 8 (𝐹𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋)
40 fvsng 6488 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
41403adant3 1101 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
4239, 41syl5eq 2697 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹𝑋) = 𝑌)
4342sneqd 4222 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {(𝐹𝑋)} = {𝑌})
4443neeq1d 2882 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({(𝐹𝑋)} ≠ {𝑍} ↔ {𝑌} ≠ {𝑍}))
45 sneqbg 4406 . . . . . . 7 (𝑌𝑊 → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
46453ad2ant2 1103 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
4746necon3abid 2859 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({𝑌} ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
4838, 44, 473bitrd 294 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
4948ifbid 4141 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(¬ 𝑌 = 𝑍, {𝑋}, ∅))
50 ifnot 4166 . . 3 if(¬ 𝑌 = 𝑍, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋})
5149, 50syl6eq 2701 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋}))
5214, 26, 513eqtrd 2689 1 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  {crab 2945  Vcvv 3231  c0 3948  ifcif 4119  {csn 4210  cop 4216  dom cdm 5143  cima 5146   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690   supp csupp 7340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-supp 7341
This theorem is referenced by: (None)
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