Theorem frinxp 5702

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 3909	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝐴 → (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝐴))
2		ssel 3909	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝐴 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝐴))
3	1, 2	anim12d 615	. . . . . . . . . 10 ⊢ (𝑧 ⊆ 𝐴 → ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
4		brinxp 5698	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5	4	ancoms 459	. . . . . . . . . 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 456	. . . . . . . 8 ⊢ (((𝑧 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑧) ∧ 𝑦 ∈ 𝑧) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
8	7	notbid 319	. . . . . . 7 ⊢ (((𝑧 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑧) ∧ 𝑦 ∈ 𝑧) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
9	8	ralbidva 3160	. . . . . 6 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑧) → (∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
10	9	rexbidva 3161	. . . . 5 ⊢ (𝑧 ⊆ 𝐴 → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
11	10	adantr 481	. . . 4 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
12	11	pm5.74i 272	. . 3 ⊢ (((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥) ↔ ((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
13	12	albii 1826	. 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 5572	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥))
15		df-fr 5572	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
16	13, 14, 15	3bitr4i 304	1 ⊢ (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063   ∩ cin 3882   ⊆ wss 3883  ∅c0 4262   class class class wbr 5073   Fr wfr 5569   × cxp 5617

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-ext 2711  ax-sep 5219  ax-pr 5363

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-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-fr 5572  df-xp 5625

This theorem is referenced by:  weinxp  5704