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Theorem suppsnop 7469
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 6330 . . . . . . 7 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
2 f1of 6290 . . . . . . 7 ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
31, 2syl 17 . . . . . 6 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
433adant3 1126 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
5 suppsnop.f . . . . . 6 𝐹 = {⟨𝑋, 𝑌⟩}
65feq1i 6189 . . . . 5 (𝐹:{𝑋}⟶{𝑌} ↔ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
74, 6sylibr 224 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹:{𝑋}⟶{𝑌})
8 snex 5049 . . . . 5 {𝑋} ∈ V
98a1i 11 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {𝑋} ∈ V)
10 fex 6645 . . . 4 ((𝐹:{𝑋}⟶{𝑌} ∧ {𝑋} ∈ V) → 𝐹 ∈ V)
117, 9, 10syl2anc 696 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 ∈ V)
12 simp3 1132 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝑍𝑈)
13 suppval 7457 . . 3 ((𝐹 ∈ V ∧ 𝑍𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
1411, 12, 13syl2anc 696 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
155a1i 11 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 = {⟨𝑋, 𝑌⟩})
1615dmeqd 5473 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom 𝐹 = dom {⟨𝑋, 𝑌⟩})
17 dmsnopg 5757 . . . . . 6 (𝑌𝑊 → dom {⟨𝑋, 𝑌⟩} = {𝑋})
18173ad2ant2 1128 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom {⟨𝑋, 𝑌⟩} = {𝑋})
1916, 18eqtrd 2786 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom 𝐹 = {𝑋})
20 rabeq 3324 . . . 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 4323 . . . . . 6 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2322imaeq2d 5616 . . . . 5 (𝑥 = 𝑋 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑋}))
2423neeq1d 2983 . . . 4 (𝑥 = 𝑋 → ((𝐹 “ {𝑥}) ≠ {𝑍} ↔ (𝐹 “ {𝑋}) ≠ {𝑍}))
2524rabsnif 4394 . . 3 {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅)
2621, 25syl6eq 2802 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅))
27 fnsng 6091 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
28273adant3 1126 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
295eqcomi 2761 . . . . . . . . . 10 {⟨𝑋, 𝑌⟩} = 𝐹
3029a1i 11 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩} = 𝐹)
3130fneq1d 6134 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({⟨𝑋, 𝑌⟩} Fn {𝑋} ↔ 𝐹 Fn {𝑋}))
3228, 31mpbid 222 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 Fn {𝑋})
33 snidg 4343 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
34333ad2ant1 1127 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝑋 ∈ {𝑋})
35 fnsnfv 6412 . . . . . . . 8 ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → {(𝐹𝑋)} = (𝐹 “ {𝑋}))
3635eqcomd 2758 . . . . . . 7 ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → (𝐹 “ {𝑋}) = {(𝐹𝑋)})
3732, 34, 36syl2anc 696 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 “ {𝑋}) = {(𝐹𝑋)})
3837neeq1d 2983 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ {(𝐹𝑋)} ≠ {𝑍}))
395fveq1i 6345 . . . . . . . 8 (𝐹𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋)
40 fvsng 6603 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
41403adant3 1126 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
4239, 41syl5eq 2798 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹𝑋) = 𝑌)
4342sneqd 4325 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {(𝐹𝑋)} = {𝑌})
4443neeq1d 2983 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({(𝐹𝑋)} ≠ {𝑍} ↔ {𝑌} ≠ {𝑍}))
45 sneqbg 4511 . . . . . . 7 (𝑌𝑊 → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
46453ad2ant2 1128 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
4746necon3abid 2960 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({𝑌} ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
4838, 44, 473bitrd 294 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
4948ifbid 4244 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(¬ 𝑌 = 𝑍, {𝑋}, ∅))
50 ifnot 4269 . . 3 if(¬ 𝑌 = 𝑍, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋})
5149, 50syl6eq 2802 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋}))
5214, 26, 513eqtrd 2790 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 1072   = wceq 1624  wcel 2131  wne 2924  {crab 3046  Vcvv 3332  c0 4050  ifcif 4222  {csn 4313  cop 4319  dom cdm 5258  cima 5261   Fn wfn 6036  wf 6037  1-1-ontowf1o 6040  cfv 6041  (class class class)co 6805   supp csupp 7455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-supp 7456
This theorem is referenced by: (None)
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