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Theorem suppsnop 7254
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 6134 . . . . . . 7 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
2 f1of 6094 . . . . . . 7 ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
31, 2syl 17 . . . . . 6 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
433adant3 1079 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
5 suppsnop.f . . . . . 6 𝐹 = {⟨𝑋, 𝑌⟩}
65feq1i 5993 . . . . 5 (𝐹:{𝑋}⟶{𝑌} ↔ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
74, 6sylibr 224 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹:{𝑋}⟶{𝑌})
8 snex 4869 . . . . 5 {𝑋} ∈ V
98a1i 11 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {𝑋} ∈ V)
10 fex 6444 . . . 4 ((𝐹:{𝑋}⟶{𝑌} ∧ {𝑋} ∈ V) → 𝐹 ∈ V)
117, 9, 10syl2anc 692 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 ∈ V)
12 simp3 1061 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝑍𝑈)
13 suppval 7242 . . 3 ((𝐹 ∈ V ∧ 𝑍𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
1411, 12, 13syl2anc 692 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
155a1i 11 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 = {⟨𝑋, 𝑌⟩})
1615dmeqd 5286 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom 𝐹 = dom {⟨𝑋, 𝑌⟩})
17 dmsnopg 5565 . . . . . 6 (𝑌𝑊 → dom {⟨𝑋, 𝑌⟩} = {𝑋})
18173ad2ant2 1081 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom {⟨𝑋, 𝑌⟩} = {𝑋})
1916, 18eqtrd 2655 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom 𝐹 = {𝑋})
20 rabeq 3179 . . . 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 4158 . . . . . 6 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2322imaeq2d 5425 . . . . 5 (𝑥 = 𝑋 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑋}))
2423neeq1d 2849 . . . 4 (𝑥 = 𝑋 → ((𝐹 “ {𝑥}) ≠ {𝑍} ↔ (𝐹 “ {𝑋}) ≠ {𝑍}))
2524rabsnif 4228 . . 3 {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅)
2621, 25syl6eq 2671 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅))
27 fnsng 5896 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
28273adant3 1079 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
295eqcomi 2630 . . . . . . . . . 10 {⟨𝑋, 𝑌⟩} = 𝐹
3029a1i 11 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} = 𝐹)
3130fneq1d 5939 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({⟨𝑋, 𝑌⟩} Fn {𝑋} ↔ 𝐹 Fn {𝑋}))
3228, 31mpbid 222 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 Fn {𝑋})
33 snidg 4177 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
34333ad2ant1 1080 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝑋 ∈ {𝑋})
35 fnsnfv 6215 . . . . . . . 8 ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → {(𝐹𝑋)} = (𝐹 “ {𝑋}))
3635eqcomd 2627 . . . . . . 7 ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → (𝐹 “ {𝑋}) = {(𝐹𝑋)})
3732, 34, 36syl2anc 692 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 “ {𝑋}) = {(𝐹𝑋)})
3837neeq1d 2849 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ {(𝐹𝑋)} ≠ {𝑍}))
395fveq1i 6149 . . . . . . . 8 (𝐹𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋)
40 fvsng 6401 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
41403adant3 1079 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
4239, 41syl5eq 2667 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹𝑋) = 𝑌)
4342sneqd 4160 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {(𝐹𝑋)} = {𝑌})
4443neeq1d 2849 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({(𝐹𝑋)} ≠ {𝑍} ↔ {𝑌} ≠ {𝑍}))
45 sneqbg 4342 . . . . . . 7 (𝑌𝑊 → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
46453ad2ant2 1081 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
4746necon3abid 2826 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({𝑌} ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
4838, 44, 473bitrd 294 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
4948ifbid 4080 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(¬ 𝑌 = 𝑍, {𝑋}, ∅))
50 ifnot 4105 . . 3 if(¬ 𝑌 = 𝑍, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋})
5149, 50syl6eq 2671 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋}))
5214, 26, 513eqtrd 2659 1 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  {crab 2911  Vcvv 3186  c0 3891  ifcif 4058  {csn 4148  cop 4154  dom cdm 5074  cima 5077   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604   supp csupp 7240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-supp 7241
This theorem is referenced by: (None)
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