Theorem phidisjnn 4616

Description: The phi operation applied to a set disjoint from the naturals has no effect. (Contributed by SF, 20-Feb-2015.)

Assertion

Ref	Expression
phidisjnn	⊢ ((A ∩ Nn ) = ∅ → Phi A = A)

Proof of Theorem phidisjnn

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

Step	Hyp	Ref	Expression
1		disj 3592	. . . . . . . . . 10 ⊢ ((A ∩ Nn ) = ∅ ↔ ∀y ∈ A ¬ y ∈ Nn )
2	1	biimpi 186	. . . . . . . . 9 ⊢ ((A ∩ Nn ) = ∅ → ∀y ∈ A ¬ y ∈ Nn )
3	2	r19.21bi 2713	. . . . . . . 8 ⊢ (((A ∩ Nn ) = ∅ ∧ y ∈ A) → ¬ y ∈ Nn )
4		iffalse 3670	. . . . . . . 8 ⊢ (¬ y ∈ Nn → if(y ∈ Nn , (y +c 1c), y) = y)
5	3, 4	syl 15	. . . . . . 7 ⊢ (((A ∩ Nn ) = ∅ ∧ y ∈ A) → if(y ∈ Nn , (y +c 1c), y) = y)
6	5	eqeq2d 2364	. . . . . 6 ⊢ (((A ∩ Nn ) = ∅ ∧ y ∈ A) → (x = if(y ∈ Nn , (y +c 1c), y) ↔ x = y))
7		equcom 1680	. . . . . 6 ⊢ (y = x ↔ x = y)
8	6, 7	syl6bbr 254	. . . . 5 ⊢ (((A ∩ Nn ) = ∅ ∧ y ∈ A) → (x = if(y ∈ Nn , (y +c 1c), y) ↔ y = x))
9	8	rexbidva 2632	. . . 4 ⊢ ((A ∩ Nn ) = ∅ → (∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y) ↔ ∃y ∈ A y = x))
10		risset 2662	. . . 4 ⊢ (x ∈ A ↔ ∃y ∈ A y = x)
11	9, 10	syl6bbr 254	. . 3 ⊢ ((A ∩ Nn ) = ∅ → (∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y) ↔ x ∈ A))
12	11	alrimiv 1631	. 2 ⊢ ((A ∩ Nn ) = ∅ → ∀x(∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y) ↔ x ∈ A))
13		df-phi 4566	. . . 4 ⊢ Phi A = {x ∣ ∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y)}
14	13	eqeq1i 2360	. . 3 ⊢ ( Phi A = A ↔ {x ∣ ∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y)} = A)
15		eqabcb 2460	. . 3 ⊢ ({x ∣ ∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y)} = A ↔ ∀x(∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y) ↔ x ∈ A))
16	14, 15	bitri 240	. 2 ⊢ ( Phi A = A ↔ ∀x(∃y ∈ A x = if(y ∈ Nn , (y +c 1c), y) ↔ x ∈ A))
17	12, 16	sylibr 203	1 ⊢ ((A ∩ Nn ) = ∅ → Phi A = A)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616   ∩ cin 3209  ∅c0 3551   ifcif 3663  1_cc1c 4135   Nn cnnc 4374   +_c cplc 4376   Phi cphi 4563

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-if 3664  df-phi 4566

This theorem is used by:  phialllem2  4618