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) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
sexp2.1	⊢ (𝜑 → 𝑅 Se 𝐴)
sexp2.2	⊢ (𝜑 → 𝑆 Se 𝐵)

Assertion

Ref	Expression
sexp2	⊢ (𝜑 → 𝑇 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.

Step	Hyp	Ref	Expression
1		elxp2 5690	. . . 4 ⊢ (𝑝 ∈ (𝐴 × 𝐵) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑝 = ⟨𝑎, 𝑏⟩)
2		xpord2.1	. . . . . . . . 9 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
3	2	xpord2pred 8150	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))
4	3	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) = (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}))
5		sexp2.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 Se 𝐴)
6		setlikespec 6333	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
7	6	ancoms 464	. . . . . . . . . . . 12 ⊢ ((𝑅 Se 𝐴 ∧ 𝑎 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
8	5, 7	sylan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
9	8	adantrr 730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10		vsnex 5411	. . . . . . . . . . 11 ⊢ {𝑎} ∈ V
11	10	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → {𝑎} ∈ V)
12	9, 11	unexd 7762	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
13		sexp2.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 Se 𝐵)
14		setlikespec 6333	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
15	14	ancoms 464	. . . . . . . . . . . 12 ⊢ ((𝑆 Se 𝐵 ∧ 𝑏 ∈ 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
16	13, 15	sylan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
17	16	adantrl 729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18		vsnex 5411	. . . . . . . . . . 11 ⊢ {𝑏} ∈ V
19	18	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → {𝑏} ∈ V)
20	17, 19	unexd 7762	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
21	12, 20	xpexd 7759	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
22	21	difexd 5307	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ∈ V)
23	4, 22	eqeltrd 2866	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V)
24		predeq3 6313	. . . . . . 7 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩))
25	24	eleq1d 2851	. . . . . 6 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V ↔ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V))
26	23, 25	syl5ibrcom 250	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
27	26	rexlimdvva 3225	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
28	1, 27	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑝 ∈ (𝐴 × 𝐵) → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
29	28	ralrimiv 3159	. 2 ⊢ (𝜑 → ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
30		dfse3 6344	. 2 ⊢ (𝑇 Se (𝐴 × 𝐵) ↔ ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
31	29, 30	sylibr 237	1 ⊢ (𝜑 → 𝑇 Se (𝐴 × 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ∖ cdif 3905   ∪ cun 3906  {csn 4594  ⟨cop 4600   class class class wbr 5114  {copab 5178   Se wse 5617   × cxp 5664  Predcpred 6308  ‘cfv 6543  1^st c1st 7993  2^nd c2nd 7994

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-se 5620  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-iota 6499  df-fun 6545  df-fv 6551  df-1st 7995  df-2nd 7996

This theorem is used by:  xpord2indlem  8152  on2recsfn  8662  on2recsov  8663  noxpordse  28175