Theorem csbima12gALTVD 45323

Description: Virtual deduction proof of csbima12 6044. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbima12 6044 is csbima12gALTVD 45323 without virtual deductions and was automatically derived from csbima12gALTVD 45323.

1::	⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   )
2:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ( ⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
3:2:	⊢ (   𝐴 ∈ 𝐶   ▶    ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
4:1:	⊢ (   𝐴 ∈ 𝐶   ▶    ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵)   )
5:3,4:	⊢ (   𝐴 ∈ 𝐶   ▶    ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
6::	⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
7:6:	⊢ ∀𝑥(𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
8:1,7:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋ 𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)   )
9:5,8:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
10::	⊢ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)
11:9,10:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ( ⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)   )
qed:11:	⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋ 𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbima12gALTVD	⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))

Proof of Theorem csbima12gALTVD

Step	Hyp	Ref	Expression
1		idn1 45001	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   )
2		csbres 5947	. . . . . . . 8 ⊢ ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)
3	2	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵))
4	1, 3	e1a 45054	. . . . . 6 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
5		rneq 5891	. . . . . 6 ⊢ (⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) → ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵))
6	4, 5	e1a 45054	. . . . 5 ⊢ (   𝐴 ∈ 𝐶   ▶   ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
7		csbrn 6167	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵)
8	7	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵))
9	1, 8	e1a 45054	. . . . 5 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵)   )
10		eqeq2 2748	. . . . . 6 ⊢ (ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) → (⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) ↔ ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)))
11	10	biimpd 229	. . . . 5 ⊢ (ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) → (⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) → ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)))
12	6, 9, 11	e11 45115	. . . 4 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
13		df-ima 5644	. . . . . 6 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
14	13	ax-gen 1797	. . . . 5 ⊢ ∀𝑥(𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
15		csbeq2 3842	. . . . . 6 ⊢ (∀𝑥(𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵) → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵))
16	15	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥(𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵) → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)))
17	1, 14, 16	e10 45121	. . . 4 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)   )
18		eqeq2 2748	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) → (⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)))
19	18	biimpd 229	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) → (⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)))
20	12, 17, 19	e11 45115	. . 3 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
21		df-ima 5644	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)
22		eqeq2 2748	. . . 4 ⊢ ((⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) → (⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)))
23	22	biimprcd 250	. . 3 ⊢ (⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) → ((⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵) → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)))
24	20, 21, 23	e10 45121	. 2 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)   )
25	24	in1 44998	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ⦋csb 3837  ran crn 5632   ↾ cres 5633   “ cima 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-vd1 44997

This theorem is referenced by: (None)