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Theorem csbima12gALTVD 44917
Description: Virtual deduction proof of csbima12 6097. 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 6097 is csbima12gALTVD 44917 without virtual deductions and was automatically derived from csbima12gALTVD 44917.
1:: (   𝐴𝐶   ▶   𝐴𝐶   )
2:1: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = ( 𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
3:2: (   𝐴𝐶   ▶    ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
4:1: (   𝐴𝐶   ▶    𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵)   )
5:3,4: (   𝐴𝐶   ▶    𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
6:: (𝐹𝐵) = ran (𝐹𝐵)
7:6: 𝑥(𝐹𝐵) = ran (𝐹𝐵)
8:1,7: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵)   )
9:5,8: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
10:: (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
11:9,10: (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = ( 𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
qed:11: (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = ( 𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
(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 44594 . . . . . . 7 (   𝐴𝐶   ▶   𝐴𝐶   )
2 csbres 6000 . . . . . . . 8 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
32a1i 11 . . . . . . 7 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
41, 3e1a 44647 . . . . . 6 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
5 rneq 5947 . . . . . 6 (𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
64, 5e1a 44647 . . . . 5 (   𝐴𝐶   ▶   ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
7 csbrn 6223 . . . . . . 7 𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵)
87a1i 11 . . . . . 6 (𝐴𝐶𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵))
91, 8e1a 44647 . . . . 5 (   𝐴𝐶   ▶   𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵)   )
10 eqeq2 2749 . . . . . 6 (ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵) ↔ 𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
1110biimpd 229 . . . . 5 (ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵) → 𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
126, 9, 11e11 44708 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
13 df-ima 5698 . . . . . 6 (𝐹𝐵) = ran (𝐹𝐵)
1413ax-gen 1795 . . . . 5 𝑥(𝐹𝐵) = ran (𝐹𝐵)
15 csbeq2 3904 . . . . . 6 (∀𝑥(𝐹𝐵) = ran (𝐹𝐵) → 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵))
1615a1i 11 . . . . 5 (𝐴𝐶 → (∀𝑥(𝐹𝐵) = ran (𝐹𝐵) → 𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵)))
171, 14, 16e10 44714 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵)   )
18 eqeq2 2749 . . . . 5 (𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
1918biimpd 229 . . . 4 (𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵) → 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
2012, 17, 19e11 44708 . . 3 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
21 df-ima 5698 . . 3 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
22 eqeq2 2749 . . . 4 ((𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
2322biimprcd 250 . . 3 (𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → ((𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
2420, 21, 23e10 44714 . 2 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
2524in1 44591 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wcel 2108  csb 3899  ran crn 5686  cres 5687  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-vd1 44590
This theorem is referenced by: (None)
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