Theorem frinxp 5768

Description: Intersection of well-founded relation with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Assertion

Ref	Expression
frinxp	⊢ (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)

Proof of Theorem frinxp

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3977	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝐴 → (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝐴))
2		ssel 3977	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝐴 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝐴))
3	1, 2	anim12d 609	. . . . . . . . . 10 ⊢ (𝑧 ⊆ 𝐴 → ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
4		brinxp 5764	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5	4	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
6	3, 5	syl6 35	. . . . . . . . 9 ⊢ (𝑧 ⊆ 𝐴 → ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
7	6	impl 455	. . . . . . . 8 ⊢ (((𝑧 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑧) ∧ 𝑦 ∈ 𝑧) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
8	7	notbid 318	. . . . . . 7 ⊢ (((𝑧 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑧) ∧ 𝑦 ∈ 𝑧) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
9	8	ralbidva 3176	. . . . . 6 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑧) → (∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
10	9	rexbidva 3177	. . . . 5 ⊢ (𝑧 ⊆ 𝐴 → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
11	10	adantr 480	. . . 4 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
12	11	pm5.74i 271	. . 3 ⊢ (((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥) ↔ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
13	12	albii 1819	. 2 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥) ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
14		df-fr 5637	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥))
15		df-fr 5637	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
16	13, 14, 15	3bitr4i 303	1 ⊢ (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143   Fr wfr 5634   × cxp 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-fr 5637  df-xp 5691

This theorem is referenced by:  weinxp  5770