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Theorem csbima12gALTVD 41590
Description: Virtual deduction proof of csbima12 5918. 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 5918 is csbima12gALTVD 41590 without virtual deductions and was automatically derived from csbima12gALTVD 41590.
 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 41267 . . . . . . 7 (   𝐴𝐶   ▶   𝐴𝐶   )
2 csbres 5825 . . . . . . . 8 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
32a1i 11 . . . . . . 7 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
41, 3e1a 41320 . . . . . 6 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
5 rneq 5774 . . . . . 6 (𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
64, 5e1a 41320 . . . . 5 (   𝐴𝐶   ▶   ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
7 csbrn 6031 . . . . . . 7 𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵)
87a1i 11 . . . . . 6 (𝐴𝐶𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵))
91, 8e1a 41320 . . . . 5 (   𝐴𝐶   ▶   𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵)   )
10 eqeq2 2813 . . . . . 6 (ran 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥ran (𝐹𝐵) = ran 𝐴 / 𝑥(𝐹𝐵) ↔ 𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
1110biimpd 232 . . . . 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 41381 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
13 df-ima 5536 . . . . . 6 (𝐹𝐵) = ran (𝐹𝐵)
1413ax-gen 1797 . . . . 5 𝑥(𝐹𝐵) = ran (𝐹𝐵)
15 csbeq2 3836 . . . . . 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 41387 . . . 4 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵)   )
18 eqeq2 2813 . . . . 5 (𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
1918biimpd 232 . . . 4 (𝐴 / 𝑥ran (𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥ran (𝐹𝐵) → 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
2012, 17, 19e11 41381 . . 3 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
21 df-ima 5536 . . 3 (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)
22 eqeq2 2813 . . . 4 ((𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → (𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
2322biimprcd 253 . . 3 (𝐴 / 𝑥(𝐹𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → ((𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) = ran (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵) → 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
2420, 21, 23e10 41387 . 2 (   𝐴𝐶   ▶   𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)   )
2524in1 41264 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538   ∈ wcel 2112  ⦋csb 3831  ran crn 5524   ↾ cres 5525   “ cima 5526 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-vd1 41263 This theorem is referenced by: (None)
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