Theorem sexp2 33793

Description: Condition for the relationship in frxp2 33791 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 5613	. . . 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 33792	. . . . . . . 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 482	. . . . . . 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 6228	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
7	6	ancoms 459	. . . . . . . . . . . 12 ⊢ ((𝑅 Se 𝐴 ∧ 𝑎 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
8	5, 7	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
9	8	adantrr 714	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10		snex 5354	. . . . . . . . . . 11 ⊢ {𝑎} ∈ V
11	10	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → {𝑎} ∈ V)
12	9, 11	unexd 7604	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
13		sexp2.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 Se 𝐵)
14		setlikespec 6228	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
15	14	ancoms 459	. . . . . . . . . . . 12 ⊢ ((𝑆 Se 𝐵 ∧ 𝑏 ∈ 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
16	13, 15	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
17	16	adantrl 713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18		snex 5354	. . . . . . . . . . 11 ⊢ {𝑏} ∈ V
19	18	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → {𝑏} ∈ V)
20	17, 19	unexd 7604	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
21	12, 20	xpexd 7601	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
22	21	difexd 5253	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ∈ V)
23	4, 22	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V)
24		predeq3 6206	. . . . . . 7 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩))
25	24	eleq1d 2823	. . . . . 6 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V ↔ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V))
26	23, 25	syl5ibrcom 246	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
27	26	rexlimdvva 3223	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
28	1, 27	syl5bi 241	. . 3 ⊢ (𝜑 → (𝑝 ∈ (𝐴 × 𝐵) → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
29	28	ralrimiv 3102	. 2 ⊢ (𝜑 → ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
30		dfse3 6239	. 2 ⊢ (𝑇 Se (𝐴 × 𝐵) ↔ ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
31	29, 30	sylibr 233	1 ⊢ (𝜑 → 𝑇 Se (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885  {csn 4561  ⟨cop 4567   class class class wbr 5074  {copab 5136   Se wse 5542   × cxp 5587  Predcpred 6201  ‘cfv 6433  1^st c1st 7829  2^nd c2nd 7830

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-se 5545  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831  df-2nd 7832

This theorem is referenced by:  xpord2ind  33794  on2recsfn  33826  on2recsov  33827  noxpordse  34109