Theorem csbrng 4967

Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

Ref	Expression
csbrng	⊢ (A ∈ V → [A / x]ran B = ran [A / x]B)

Proof of Theorem csbrng

Dummy variables w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbabg 3198	. . 3 ⊢ (A ∈ V → [A / x]{y ∣ ∃w⟨w, y⟩ ∈ B} = {y ∣ [̣A / x]̣∃w⟨w, y⟩ ∈ B})
2		sbcexg 3097	. . . . 5 ⊢ (A ∈ V → ([̣A / x]̣∃w⟨w, y⟩ ∈ B ↔ ∃w[̣A / x]̣⟨w, y⟩ ∈ B))
3		sbcel2g 3158	. . . . . 6 ⊢ (A ∈ V → ([̣A / x]̣⟨w, y⟩ ∈ B ↔ ⟨w, y⟩ ∈ [A / x]B))
4	3	exbidv 1626	. . . . 5 ⊢ (A ∈ V → (∃w[̣A / x]̣⟨w, y⟩ ∈ B ↔ ∃w⟨w, y⟩ ∈ [A / x]B))
5	2, 4	bitrd 244	. . . 4 ⊢ (A ∈ V → ([̣A / x]̣∃w⟨w, y⟩ ∈ B ↔ ∃w⟨w, y⟩ ∈ [A / x]B))
6	5	abbidv 2468	. . 3 ⊢ (A ∈ V → {y ∣ [̣A / x]̣∃w⟨w, y⟩ ∈ B} = {y ∣ ∃w⟨w, y⟩ ∈ [A / x]B})
7	1, 6	eqtrd 2385	. 2 ⊢ (A ∈ V → [A / x]{y ∣ ∃w⟨w, y⟩ ∈ B} = {y ∣ ∃w⟨w, y⟩ ∈ [A / x]B})
8		dfrn3 4904	. . 3 ⊢ ran B = {y ∣ ∃w⟨w, y⟩ ∈ B}
9	8	csbeq2i 3163	. 2 ⊢ [A / x]ran B = [A / x]{y ∣ ∃w⟨w, y⟩ ∈ B}
10		dfrn3 4904	. 2 ⊢ ran [A / x]B = {y ∣ ∃w⟨w, y⟩ ∈ [A / x]B}
11	7, 9, 10	3eqtr4g 2410	1 ⊢ (A ∈ V → [A / x]ran B = ran [A / x]B)

Syntax hints:   → wi 4  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3047  [csb 3137  ⟨cop 4562  ran crn 4774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-br 4641  df-ima 4728  df-rn 4787

This theorem is referenced by: (None)