Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  csbima12gALTVD Structured version   Visualization version   GIF version

Theorem csbima12gALTVD 39447
Description: Virtual deduction proof of csbima12 5518. 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 5518 is csbima12gALTVD 39447 without virtual deductions and was automatically derived from csbima12gALTVD 39447.
 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 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

Proof of Theorem csbima12gALTVD
StepHypRef Expression
1 idn1 39107 . . . . . . 7 (   𝐴𝐶   ▶   𝐴𝐶   )
2 csbresgOLD 39370 . . . . . . 7 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
31, 2e1a 39169 . . . . . 6 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
4 rneq 5383 . . . . . 6 (𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
53, 4e1a 39169 . . . . 5 (   𝐴𝐶   ▶   ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
6 csbrngOLD 39371 . . . . . 6 (𝐴𝐶𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵))
71, 6e1a 39169 . . . . 5 (   𝐴𝐶   ▶   𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵)   )
8 eqeq2 2662 . . . . . 6 (ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵) ↔ 𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
98biimpd 219 . . . . 5 (ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵) → 𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
105, 7, 9e11 39230 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
11 df-ima 5156 . . . . . 6 (𝐹𝐵) = ran (𝐹𝐵)
1211ax-gen 1762 . . . . 5 𝑥(𝐹𝐵) = ran (𝐹𝐵)
13 csbeq2gOLD 39082 . . . . 5 (𝐴𝐶 → (∀𝑥(𝐹𝐵) = ran (𝐹𝐵) → 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵)))
141, 12, 13e10 39236 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵)   )
15 eqeq2 2662 . . . . 5 (𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
1615biimpd 219 . . . 4 (𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵) → 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
1710, 14, 16e11 39230 . . 3 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
18 df-ima 5156 . . 3 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
19 eqeq2 2662 . . . 4 ((𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
2019biimprcd 240 . . 3 (𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → ((𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
2117, 18, 20e10 39236 . 2 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
2221in1 39104 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521   = wceq 1523   ∈ wcel 2030  ⦋csb 3566  ran crn 5144   ↾ cres 5145   “ cima 5146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-vd1 39103 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator