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Theorem sexp2 8172
Description: Condition for the relation in frxp2 8170 to be set-like. (Contributed by Scott Fenton, 19-Aug-2024.)
Hypotheses
Ref Expression
xpord2.1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st𝑥)𝑅(1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥)𝑆(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}
sexp2.1 (𝜑𝑅 Se 𝐴)
sexp2.2 (𝜑𝑆 Se 𝐵)
Assertion
Ref Expression
sexp2 (𝜑𝑇 Se (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑇(𝑥,𝑦)

Proof of Theorem sexp2
Dummy variables 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 5708 . . . 4 (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑎𝐴𝑏𝐵 𝑝 = ⟨𝑎, 𝑏⟩)
2 xpord2.1 . . . . . . . . 9 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st𝑥)𝑅(1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥)𝑆(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}
32xpord2pred 8171 . . . . . . . 8 ((𝑎𝐴𝑏𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))
43adantl 481 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))
5 sexp2.1 . . . . . . . . . . . 12 (𝜑𝑅 Se 𝐴)
6 setlikespec 6345 . . . . . . . . . . . . 13 ((𝑎𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
76ancoms 458 . . . . . . . . . . . 12 ((𝑅 Se 𝐴𝑎𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
85, 7sylan 580 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
98adantrr 717 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10 vsnex 5433 . . . . . . . . . . 11 {𝑎} ∈ V
1110a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑎} ∈ V)
129, 11unexd 7775 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
13 sexp2.2 . . . . . . . . . . . 12 (𝜑𝑆 Se 𝐵)
14 setlikespec 6345 . . . . . . . . . . . . 13 ((𝑏𝐵𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1514ancoms 458 . . . . . . . . . . . 12 ((𝑆 Se 𝐵𝑏𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1613, 15sylan 580 . . . . . . . . . . 11 ((𝜑𝑏𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1716adantrl 716 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18 vsnex 5433 . . . . . . . . . . 11 {𝑏} ∈ V
1918a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑏} ∈ V)
2017, 19unexd 7775 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
2112, 20xpexd 7772 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
2221difexd 5330 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ∈ V)
234, 22eqeltrd 2840 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V)
24 predeq3 6324 . . . . . . 7 (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩))
2524eleq1d 2825 . . . . . 6 (𝑝 = ⟨𝑎, 𝑏⟩ → (Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V ↔ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V))
2623, 25syl5ibrcom 247 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
2726rexlimdvva 3212 . . . 4 (𝜑 → (∃𝑎𝐴𝑏𝐵 𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
281, 27biimtrid 242 . . 3 (𝜑 → (𝑝 ∈ (𝐴 × 𝐵) → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
2928ralrimiv 3144 . 2 (𝜑 → ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
30 dfse3 6356 . 2 (𝑇 Se (𝐴 × 𝐵) ↔ ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
3129, 30sylibr 234 1 (𝜑𝑇 Se (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  cun 3948  {csn 4625  cop 4631   class class class wbr 5142  {copab 5204   Se wse 5634   × cxp 5682  Predcpred 6319  cfv 6560  1st c1st 8013  2nd c2nd 8014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-se 5637  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fun 6562  df-fv 6568  df-1st 8015  df-2nd 8016
This theorem is referenced by:  xpord2indlem  8173  on2recsfn  8706  on2recsov  8707  noxpordse  27986
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