Theorem clos1is 5882

Description: Induction scheme for closures. Hypotheses one through three set up existence properties, hypothesis four sets up stratification, hypotheses five through seven set up implicit substitution, and hypotheses eight and nine set up the base and induction steps. (Contributed by SF, 13-Feb-2015.)

Hypotheses

Ref	Expression
clos1is.1	⊢ S ∈ V
clos1is.2	⊢ R ∈ V
clos1is.3	⊢ C = Clos1 (S, R)
clos1is.4	⊢ {x ∣ φ} ∈ V
clos1is.5	⊢ (x = y → (φ ↔ ψ))
clos1is.6	⊢ (x = z → (φ ↔ χ))
clos1is.7	⊢ (x = A → (φ ↔ θ))
clos1is.8	⊢ (x ∈ S → φ)
clos1is.9	⊢ ((y ∈ C ∧ yRz ∧ ψ) → χ)

Assertion

Ref	Expression
clos1is	⊢ (A ∈ C → θ)

Distinct variable groups:   x,A   y,C,z   χ,x   φ,y,z   ψ,x   y,R,z   x,S   θ,x   x,y,z

Allowed substitution hints:   φ(x)   ψ(y,z)   χ(y,z)   θ(y,z)   A(y,z)   C(x)   R(x)   S(y,z)

Proof of Theorem clos1is

Step	Hyp	Ref	Expression
1		clos1is.4	. . . 4 ⊢ {x ∣ φ} ∈ V
2		ssab 3337	. . . . 5 ⊢ (S ⊆ {x ∣ φ} ↔ ∀x(x ∈ S → φ))
3		clos1is.8	. . . . 5 ⊢ (x ∈ S → φ)
4	2, 3	mpgbir 1550	. . . 4 ⊢ S ⊆ {x ∣ φ}
5		clos1is.9	. . . . . . . 8 ⊢ ((y ∈ C ∧ yRz ∧ ψ) → χ)
6	5	3expib 1154	. . . . . . 7 ⊢ (y ∈ C → ((yRz ∧ ψ) → χ))
7		vex 2863	. . . . . . . . . 10 ⊢ y ∈ V
8		clos1is.5	. . . . . . . . . 10 ⊢ (x = y → (φ ↔ ψ))
9	7, 8	elab 2986	. . . . . . . . 9 ⊢ (y ∈ {x ∣ φ} ↔ ψ)
10	9	anbi1i 676	. . . . . . . 8 ⊢ ((y ∈ {x ∣ φ} ∧ yRz) ↔ (ψ ∧ yRz))
11		ancom 437	. . . . . . . 8 ⊢ ((ψ ∧ yRz) ↔ (yRz ∧ ψ))
12	10, 11	bitri 240	. . . . . . 7 ⊢ ((y ∈ {x ∣ φ} ∧ yRz) ↔ (yRz ∧ ψ))
13		vex 2863	. . . . . . . 8 ⊢ z ∈ V
14		clos1is.6	. . . . . . . 8 ⊢ (x = z → (φ ↔ χ))
15	13, 14	elab 2986	. . . . . . 7 ⊢ (z ∈ {x ∣ φ} ↔ χ)
16	6, 12, 15	3imtr4g 261	. . . . . 6 ⊢ (y ∈ C → ((y ∈ {x ∣ φ} ∧ yRz) → z ∈ {x ∣ φ}))
17	16	alrimiv 1631	. . . . 5 ⊢ (y ∈ C → ∀z((y ∈ {x ∣ φ} ∧ yRz) → z ∈ {x ∣ φ}))
18	17	rgen 2680	. . . 4 ⊢ ∀y ∈ C ∀z((y ∈ {x ∣ φ} ∧ yRz) → z ∈ {x ∣ φ})
19		clos1is.1	. . . . 5 ⊢ S ∈ V
20		clos1is.2	. . . . 5 ⊢ R ∈ V
21		clos1is.3	. . . . 5 ⊢ C = Clos1 (S, R)
22	19, 20, 21	clos1induct 5881	. . . 4 ⊢ (({x ∣ φ} ∈ V ∧ S ⊆ {x ∣ φ} ∧ ∀y ∈ C ∀z((y ∈ {x ∣ φ} ∧ yRz) → z ∈ {x ∣ φ})) → C ⊆ {x ∣ φ})
23	1, 4, 18, 22	mp3an 1277	. . 3 ⊢ C ⊆ {x ∣ φ}
24	23	sseli 3270	. 2 ⊢ (A ∈ C → A ∈ {x ∣ φ})
25		clos1is.7	. . 3 ⊢ (x = A → (φ ↔ θ))
26	25	elabg 2987	. 2 ⊢ (A ∈ C → (A ∈ {x ∣ φ} ↔ θ))
27	24, 26	mpbid 201	1 ⊢ (A ∈ C → θ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  Vcvv 2860   ⊆ wss 3258   class class class wbr 4640   Clos1 cclos1 5873

This theorem was proved from 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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-2nd 4798  df-txp 5737  df-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-clos1 5874

This theorem is referenced by:  clos1basesuc  5883