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Theorem sexp2 8151
Description: Condition for the relation in frxp2 8149 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 5702 . . . 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 8150 . . . . . . . 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 6331 . . . . . . . . . . . . 13 ((𝑎𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
76ancoms 458 . . . . . . . . . . . 12 ((𝑅 Se 𝐴𝑎𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
85, 7sylan 579 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
98adantrr 716 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10 vsnex 5431 . . . . . . . . . . 11 {𝑎} ∈ V
1110a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑎} ∈ V)
129, 11unexd 7756 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
13 sexp2.2 . . . . . . . . . . . 12 (𝜑𝑆 Se 𝐵)
14 setlikespec 6331 . . . . . . . . . . . . 13 ((𝑏𝐵𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1514ancoms 458 . . . . . . . . . . . 12 ((𝑆 Se 𝐵𝑏𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1613, 15sylan 579 . . . . . . . . . . 11 ((𝜑𝑏𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1716adantrl 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18 vsnex 5431 . . . . . . . . . . 11 {𝑏} ∈ V
1918a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑏} ∈ V)
2017, 19unexd 7756 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
2112, 20xpexd 7753 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
2221difexd 5331 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ∈ V)
234, 22eqeltrd 2829 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V)
24 predeq3 6309 . . . . . . 7 (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩))
2524eleq1d 2814 . . . . . 6 (𝑝 = ⟨𝑎, 𝑏⟩ → (Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V ↔ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V))
2623, 25syl5ibrcom 246 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
2726rexlimdvva 3208 . . . 4 (𝜑 → (∃𝑎𝐴𝑏𝐵 𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
281, 27biimtrid 241 . . 3 (𝜑 → (𝑝 ∈ (𝐴 × 𝐵) → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
2928ralrimiv 3142 . 2 (𝜑 → ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
30 dfse3 6342 . 2 (𝑇 Se (𝐴 × 𝐵) ↔ ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
3129, 30sylibr 233 1 (𝜑𝑇 Se (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  wne 2937  wral 3058  wrex 3067  Vcvv 3471  cdif 3944  cun 3945  {csn 4629  cop 4635   class class class wbr 5148  {copab 5210   Se wse 5631   × cxp 5676  Predcpred 6304  cfv 6548  1st c1st 7991  2nd c2nd 7992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-se 5634  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-iota 6500  df-fun 6550  df-fv 6556  df-1st 7993  df-2nd 7994
This theorem is referenced by:  xpord2indlem  8152  on2recsfn  8687  on2recsov  8688  noxpordse  27868
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