MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptex Structured version   Visualization version   GIF version

Theorem fvmptex 6251
Description: Express a function 𝐹 whose value 𝐵 may not always be a set in terms of another function 𝐺 for which sethood is guaranteed. (Note that ( I ‘𝐵) is just shorthand for if(𝐵 ∈ V, 𝐵, ∅), and it is always a set by fvex 6158.) Note also that these functions are not the same; wherever 𝐵(𝐶) is not a set, 𝐶 is not in the domain of 𝐹 (so it evaluates to the empty set), but 𝐶 is in the domain of 𝐺, and 𝐺(𝐶) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
fvmptex.1 𝐹 = (𝑥𝐴𝐵)
fvmptex.2 𝐺 = (𝑥𝐴 ↦ ( I ‘𝐵))
Assertion
Ref Expression
fvmptex (𝐹𝐶) = (𝐺𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem fvmptex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3517 . . . 4 (𝑦 = 𝐶𝑦 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
2 fvmptex.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
3 nfcv 2761 . . . . . 6 𝑦𝐵
4 nfcsb1v 3530 . . . . . 6 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3523 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 4709 . . . . 5 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
72, 6eqtri 2643 . . . 4 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
81, 7fvmpti 6238 . . 3 (𝐶𝐴 → (𝐹𝐶) = ( I ‘𝐶 / 𝑥𝐵))
91fveq2d 6152 . . . 4 (𝑦 = 𝐶 → ( I ‘𝑦 / 𝑥𝐵) = ( I ‘𝐶 / 𝑥𝐵))
10 fvmptex.2 . . . . 5 𝐺 = (𝑥𝐴 ↦ ( I ‘𝐵))
11 nfcv 2761 . . . . . 6 𝑦( I ‘𝐵)
12 nfcv 2761 . . . . . . 7 𝑥 I
1312, 4nffv 6155 . . . . . 6 𝑥( I ‘𝑦 / 𝑥𝐵)
145fveq2d 6152 . . . . . 6 (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘𝑦 / 𝑥𝐵))
1511, 13, 14cbvmpt 4709 . . . . 5 (𝑥𝐴 ↦ ( I ‘𝐵)) = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
1610, 15eqtri 2643 . . . 4 𝐺 = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
17 fvex 6158 . . . 4 ( I ‘𝐶 / 𝑥𝐵) ∈ V
189, 16, 17fvmpt 6239 . . 3 (𝐶𝐴 → (𝐺𝐶) = ( I ‘𝐶 / 𝑥𝐵))
198, 18eqtr4d 2658 . 2 (𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
202dmmptss 5590 . . . . . 6 dom 𝐹𝐴
2120sseli 3579 . . . . 5 (𝐶 ∈ dom 𝐹𝐶𝐴)
2221con3i 150 . . . 4 𝐶𝐴 → ¬ 𝐶 ∈ dom 𝐹)
23 ndmfv 6175 . . . 4 𝐶 ∈ dom 𝐹 → (𝐹𝐶) = ∅)
2422, 23syl 17 . . 3 𝐶𝐴 → (𝐹𝐶) = ∅)
25 fvex 6158 . . . . . 6 ( I ‘𝐵) ∈ V
2625, 10dmmpti 5980 . . . . 5 dom 𝐺 = 𝐴
2726eleq2i 2690 . . . 4 (𝐶 ∈ dom 𝐺𝐶𝐴)
28 ndmfv 6175 . . . 4 𝐶 ∈ dom 𝐺 → (𝐺𝐶) = ∅)
2927, 28sylnbir 321 . . 3 𝐶𝐴 → (𝐺𝐶) = ∅)
3024, 29eqtr4d 2658 . 2 𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
3119, 30pm2.61i 176 1 (𝐹𝐶) = (𝐺𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  csb 3514  c0 3891  cmpt 4673   I cid 4984  dom cdm 5074  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855
This theorem is referenced by:  fvmptnf  6258  sumeq2ii  14357  prodeq2ii  14568
  Copyright terms: Public domain W3C validator