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Theorem sexp2 8083
Description: Condition for the relation in frxp2 8081 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 5662 . . . 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 8082 . . . . . . . 8 ((𝑎𝐴𝑏𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))
43adantl 483 . . . . . . 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 6284 . . . . . . . . . . . . 13 ((𝑎𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
76ancoms 460 . . . . . . . . . . . 12 ((𝑅 Se 𝐴𝑎𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
85, 7sylan 581 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
98adantrr 716 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10 vsnex 5391 . . . . . . . . . . 11 {𝑎} ∈ V
1110a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑎} ∈ V)
129, 11unexd 7693 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
13 sexp2.2 . . . . . . . . . . . 12 (𝜑𝑆 Se 𝐵)
14 setlikespec 6284 . . . . . . . . . . . . 13 ((𝑏𝐵𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1514ancoms 460 . . . . . . . . . . . 12 ((𝑆 Se 𝐵𝑏𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1613, 15sylan 581 . . . . . . . . . . 11 ((𝜑𝑏𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1716adantrl 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18 vsnex 5391 . . . . . . . . . . 11 {𝑏} ∈ V
1918a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → {𝑏} ∈ V)
2017, 19unexd 7693 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
2112, 20xpexd 7690 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
2221difexd 5291 . . . . . . 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 6262 . . . . . . 7 (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩))
2524eleq1d 2823 . . . . . 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 3206 . . . 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 3143 . 2 (𝜑 → ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
30 dfse3 6295 . 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3448  cdif 3912  cun 3913  {csn 4591  cop 4597   class class class wbr 5110  {copab 5172   Se wse 5591   × cxp 5636  Predcpred 6257  cfv 6501  1st c1st 7924  2nd c2nd 7925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-se 5594  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-iota 6453  df-fun 6503  df-fv 6509  df-1st 7926  df-2nd 7927
This theorem is referenced by:  xpord2indlem  8084  on2recsfn  8618  on2recsov  8619  noxpordse  27286
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