Theorem sexp2 8093

Description: Condition for the relation in frxp2 8091 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 5649	. . . 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 8092	. . . . . . . 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 6283	. . . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
9	8	adantrr 723	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10		vsnex 5371	. . . . . . . . . . 11 ⊢ {𝑎} ∈ V
11	10	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → {𝑎} ∈ V)
12	9, 11	unexd 7704	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
13		sexp2.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 Se 𝐵)
14		setlikespec 6283	. . . . . . . . . . . . 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 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
17	16	adantrl 722	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18		vsnex 5371	. . . . . . . . . . 11 ⊢ {𝑏} ∈ V
19	18	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → {𝑏} ∈ V)
20	17, 19	unexd 7704	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
21	12, 20	xpexd 7701	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
22	21	difexd 5266	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∖ {⟨𝑎, 𝑏⟩}) ∈ V)
23	4, 22	eqeltrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V)
24		predeq3 6263	. . . . . . 7 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) = Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩))
25	24	eleq1d 2825	. . . . . 6 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V ↔ Pred(𝑇, (𝐴 × 𝐵), ⟨𝑎, 𝑏⟩) ∈ V))
26	23, 25	syl5ibrcom 248	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
27	26	rexlimdvva 3197	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑝 = ⟨𝑎, 𝑏⟩ → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
28	1, 27	biimtrid 243	. . 3 ⊢ (𝜑 → (𝑝 ∈ (𝐴 × 𝐵) → Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V))
29	28	ralrimiv 3131	. 2 ⊢ (𝜑 → ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
30		dfse3 6294	. 2 ⊢ (𝑇 Se (𝐴 × 𝐵) ↔ ∀𝑝 ∈ (𝐴 × 𝐵)Pred(𝑇, (𝐴 × 𝐵), 𝑝) ∈ V)
31	29, 30	sylibr 235	1 ⊢ (𝜑 → 𝑇 Se (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ∖ cdif 3887   ∪ cun 3888  {csn 4562  ⟨cop 4568   class class class wbr 5079  {copab 5141   Se wse 5576   × cxp 5623  Predcpred 6258  ‘cfv 6492  1^st c1st 7936  2^nd c2nd 7937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-se 5579  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7938  df-2nd 7939

This theorem is referenced by:  xpord2indlem  8094  on2recsfn  8600  on2recsov  8601  noxpordse  27969