Theorem nffrfor 4333

Description: Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nffrfor.r	⊢ Ⅎ𝑥𝑅
nffrfor.a	⊢ Ⅎ𝑥𝐴
nffrfor.s	⊢ Ⅎ𝑥𝑆

Assertion

Ref	Expression
nffrfor	⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆

Proof of Theorem nffrfor

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frfor 4316	. 2 ⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆))
2		nffrfor.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑥𝑣
4		nffrfor.r	. . . . . . . 8 ⊢ Ⅎ𝑥𝑅
5		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑥𝑢
6	3, 4, 5	nfbr 4035	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣𝑅𝑢
7		nffrfor.s	. . . . . . . 8 ⊢ Ⅎ𝑥𝑆
8	7	nfcri 2306	. . . . . . 7 ⊢ Ⅎ𝑥 𝑣 ∈ 𝑆
9	6, 8	nfim 1565	. . . . . 6 ⊢ Ⅎ𝑥(𝑣𝑅𝑢 → 𝑣 ∈ 𝑆)
10	2, 9	nfralxy 2508	. . . . 5 ⊢ Ⅎ𝑥∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆)
11	7	nfcri 2306	. . . . 5 ⊢ Ⅎ𝑥 𝑢 ∈ 𝑆
12	10, 11	nfim 1565	. . . 4 ⊢ Ⅎ𝑥(∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆)
13	2, 12	nfralxy 2508	. . 3 ⊢ Ⅎ𝑥∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆)
14	2, 7	nfss 3140	. . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑆
15	13, 14	nfim 1565	. 2 ⊢ Ⅎ𝑥(∀𝑢 ∈ 𝐴 (∀𝑣 ∈ 𝐴 (𝑣𝑅𝑢 → 𝑣 ∈ 𝑆) → 𝑢 ∈ 𝑆) → 𝐴 ⊆ 𝑆)
16	1, 15	nfxfr 1467	1 ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆

Syntax hints:   → wi 4  Ⅎwnf 1453   ∈ wcel 2141  Ⅎwnfc 2299  ∀wral 2448   ⊆ wss 3121   class class class wbr 3989   FrFor wfrfor 4312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-frfor 4316

This theorem is referenced by:  nffr  4334