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Theorem suppsnop 8163
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 6868 . . . . . . 7 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
2 f1of 6827 . . . . . . 7 ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
31, 2syl 17 . . . . . 6 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
433adant3 1129 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
5 suppsnop.f . . . . . 6 𝐹 = {⟨𝑋, 𝑌⟩}
65feq1i 6702 . . . . 5 (𝐹:{𝑋}⟶{𝑌} ↔ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
74, 6sylibr 233 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹:{𝑋}⟶{𝑌})
8 snex 5424 . . . 4 {𝑋} ∈ V
9 fex 7223 . . . 4 ((𝐹:{𝑋}⟶{𝑌} ∧ {𝑋} ∈ V) → 𝐹 ∈ V)
107, 8, 9sylancl 585 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 ∈ V)
11 simp3 1135 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝑍𝑈)
12 suppval 8148 . . 3 ((𝐹 ∈ V ∧ 𝑍𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
1310, 11, 12syl2anc 583 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 supp 𝑍) = {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
147fdmd 6722 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → dom 𝐹 = {𝑋})
1514rabeqdv 3441 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}})
16 sneq 4633 . . . . . 6 (𝑥 = 𝑋 → {𝑥} = {𝑋})
1716imaeq2d 6053 . . . . 5 (𝑥 = 𝑋 → (𝐹 “ {𝑥}) = (𝐹 “ {𝑋}))
1817neeq1d 2994 . . . 4 (𝑥 = 𝑋 → ((𝐹 “ {𝑥}) ≠ {𝑍} ↔ (𝐹 “ {𝑋}) ≠ {𝑍}))
1918rabsnif 4722 . . 3 {𝑥 ∈ {𝑋} ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅)
2015, 19eqtrdi 2782 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {𝑥 ∈ dom 𝐹 ∣ (𝐹 “ {𝑥}) ≠ {𝑍}} = if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅))
217ffnd 6712 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝐹 Fn {𝑋})
22 snidg 4657 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
23223ad2ant1 1130 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → 𝑋 ∈ {𝑋})
24 fnsnfv 6964 . . . . . . . 8 ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → {(𝐹𝑋)} = (𝐹 “ {𝑋}))
2524eqcomd 2732 . . . . . . 7 ((𝐹 Fn {𝑋} ∧ 𝑋 ∈ {𝑋}) → (𝐹 “ {𝑋}) = {(𝐹𝑋)})
2621, 23, 25syl2anc 583 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 “ {𝑋}) = {(𝐹𝑋)})
2726neeq1d 2994 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ {(𝐹𝑋)} ≠ {𝑍}))
285fveq1i 6886 . . . . . . . 8 (𝐹𝑋) = ({⟨𝑋, 𝑌⟩}‘𝑋)
29 fvsng 7174 . . . . . . . . 9 ((𝑋𝑉𝑌𝑊) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
30293adant3 1129 . . . . . . . 8 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({⟨𝑋, 𝑌⟩}‘𝑋) = 𝑌)
3128, 30eqtrid 2778 . . . . . . 7 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹𝑋) = 𝑌)
3231sneqd 4635 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → {(𝐹𝑋)} = {𝑌})
3332neeq1d 2994 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({(𝐹𝑋)} ≠ {𝑍} ↔ {𝑌} ≠ {𝑍}))
34 sneqbg 4839 . . . . . . 7 (𝑌𝑊 → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
35343ad2ant2 1131 . . . . . 6 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({𝑌} = {𝑍} ↔ 𝑌 = 𝑍))
3635necon3abid 2971 . . . . 5 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ({𝑌} ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
3727, 33, 363bitrd 305 . . . 4 ((𝑋𝑉𝑌𝑊𝑍𝑈) → ((𝐹 “ {𝑋}) ≠ {𝑍} ↔ ¬ 𝑌 = 𝑍))
3837ifbid 4546 . . 3 ((𝑋𝑉𝑌𝑊𝑍𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(¬ 𝑌 = 𝑍, {𝑋}, ∅))
39 ifnot 4575 . . 3 if(¬ 𝑌 = 𝑍, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋})
4038, 39eqtrdi 2782 . 2 ((𝑋𝑉𝑌𝑊𝑍𝑈) → if((𝐹 “ {𝑋}) ≠ {𝑍}, {𝑋}, ∅) = if(𝑌 = 𝑍, ∅, {𝑋}))
4113, 20, 403eqtrd 2770 1 ((𝑋𝑉𝑌𝑊𝑍𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2934  {crab 3426  Vcvv 3468  c0 4317  ifcif 4523  {csn 4623  cop 4629  dom cdm 5669  cima 5672   Fn wfn 6532  wf 6533  1-1-ontowf1o 6536  cfv 6537  (class class class)co 7405   supp csupp 8146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-supp 8147
This theorem is referenced by: (None)
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