Theorem clos1is 5881

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 3336	. . . . 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 2862	. . . . . . . . . 10 ⊢ y ∈ V
8		clos1is.5	. . . . . . . . . 10 ⊢ (x = y → (φ ↔ ψ))
9	7, 8	elab 2985	. . . . . . . . 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 2862	. . . . . . . 8 ⊢ z ∈ V
14		clos1is.6	. . . . . . . 8 ⊢ (x = z → (φ ↔ χ))
15	13, 14	elab 2985	. . . . . . 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 2679	. . . 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 5880	. . . 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 3269	. 2 ⊢ (A ∈ C → A ∈ {x ∣ φ})
25		clos1is.7	. . 3 ⊢ (x = A → (φ ↔ θ))
26	25	elabg 2986	. 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 2614  Vcvv 2859   ⊆ wss 3257   class class class wbr 4639   Clos1 cclos1 5872

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-2nd 4797  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-clos1 5873

This theorem is referenced by:  clos1basesuc  5882