Theorem frinxp 5782

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 4002	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝐴 → (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝐴))
2		ssel 4002	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝐴 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝐴))
3	1, 2	anim12d 608	. . . . . . . . . 10 ⊢ (𝑧 ⊆ 𝐴 → ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
4		brinxp 5778	. . . . . . . . . . 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 3182	. . . . . 6 ⊢ ((𝑧 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑧) → (∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
10	9	rexbidva 3183	. . . . 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 1817	. 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 5652	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 ¬ 𝑦𝑅𝑥))
15		df-fr 5652	. 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 1535   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166   Fr wfr 5649   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-fr 5652  df-xp 5706

This theorem is referenced by:  weinxp  5784