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Theorem sexp2 8075
Description: Condition for the relation in frxp2 8073 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 5656 . . . 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 8074 . . . . . . . 8 ((𝑎𝐴𝑏𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))
43adantl 482 . . . . . . 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 6278 . . . . . . . . . . . . 13 ((𝑎𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
76ancoms 459 . . . . . . . . . . . 12 ((𝑅 Se 𝐴𝑎𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
85, 7sylan 580 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
98adantrr 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10 vsnex 5385 . . . . . . . . . . 11 {𝑎} ∈ V
1110a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑎} ∈ V)
129, 11unexd 7685 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
13 sexp2.2 . . . . . . . . . . . 12 (𝜑𝑆 Se 𝐵)
14 setlikespec 6278 . . . . . . . . . . . . 13 ((𝑏𝐵𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1514ancoms 459 . . . . . . . . . . . 12 ((𝑆 Se 𝐵𝑏𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1613, 15sylan 580 . . . . . . . . . . 11 ((𝜑𝑏𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1716adantrl 714 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18 vsnex 5385 . . . . . . . . . . 11 {𝑏} ∈ V
1918a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑏} ∈ V)
2017, 19unexd 7685 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
2112, 20xpexd 7682 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
2221difexd 5285 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ∈ V)
234, 22eqeltrd 2838 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V)
24 predeq3 6256 . . . . . . 7 (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩))
2524eleq1d 2822 . . . . . 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 3204 . . . 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 3141 . 2 (𝜑 → ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
30 dfse3 6289 . 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2942  wral 3063  wrex 3072  Vcvv 3444  cdif 3906  cun 3907  {csn 4585  cop 4591   class class class wbr 5104  {copab 5166   Se wse 5585   × cxp 5630  Predcpred 6251  cfv 6494  1st c1st 7916  2nd c2nd 7917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-se 5588  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-iota 6446  df-fun 6496  df-fv 6502  df-1st 7918  df-2nd 7919
This theorem is referenced by:  xpord2indlem  8076  on2recsfn  8610  on2recsov  8611  noxpordse  27260
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