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Theorem sexp2 8170
Description: Condition for the relation in frxp2 8168 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 5713 . . . 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 8169 . . . . . . . 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 6348 . . . . . . . . . . . . 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 5440 . . . . . . . . . . 11 {𝑎} ∈ V
1110a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑎} ∈ V)
129, 11unexd 7773 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
13 sexp2.2 . . . . . . . . . . . 12 (𝜑𝑆 Se 𝐵)
14 setlikespec 6348 . . . . . . . . . . . . 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 5440 . . . . . . . . . . 11 {𝑏} ∈ V
1918a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑏} ∈ V)
2017, 19unexd 7773 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
2112, 20xpexd 7770 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
2221difexd 5337 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ∈ V)
234, 22eqeltrd 2839 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V)
24 predeq3 6327 . . . . . . 7 (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩))
2524eleq1d 2824 . . . . . 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 3211 . . . 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 3143 . 2 (𝜑 → ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
30 dfse3 6359 . 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 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  {csn 4631  cop 4637   class class class wbr 5148  {copab 5210   Se wse 5639   × cxp 5687  Predcpred 6322  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-se 5642  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  xpord2indlem  8171  on2recsfn  8704  on2recsov  8705  noxpordse  28000
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