Theorem csbima12gALTVD 45252

Description: Virtual deduction proof of csbima12 6046. 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 6046 is csbima12gALTVD 45252 without virtual deductions and was automatically derived from csbima12gALTVD 45252.

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 44930	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   )
2		csbres 5949	. . . . . . . 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 44983	. . . . . 6 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
5		rneq 5893	. . . . . 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 44983	. . . . 5 ⊢ (   𝐴 ∈ 𝐶   ▶   ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
7		csbrn 6169	. . . . . . 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 44983	. . . . 5 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran ⦋𝐴 / 𝑥⦌(𝐹 ↾ 𝐵)   )
10		eqeq2 2749	. . . . . 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 45044	. . . 4 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
13		df-ima 5645	. . . . . 6 ⊢ (𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
14	13	ax-gen 1797	. . . . 5 ⊢ ∀𝑥(𝐹 “ 𝐵) = ran (𝐹 ↾ 𝐵)
15		csbeq2 3856	. . . . . 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 45050	. . . 4 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ⦋𝐴 / 𝑥⦌ran (𝐹 ↾ 𝐵)   )
18		eqeq2 2749	. . . . 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 45044	. . 3 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)   )
21		df-ima 5645	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵) = ran (⦋𝐴 / 𝑥⦌𝐹 ↾ ⦋𝐴 / 𝑥⦌𝐵)
22		eqeq2 2749	. . . 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 45050	. 2 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵)   )
25	24	in1 44927	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ 𝐵) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ⦋csb 3851  ran crn 5633   ↾ cres 5634   “ cima 5635

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 2709  ax-sep 5243  ax-pr 5379

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 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-vd1 44926

This theorem is referenced by: (None)