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Theorem fvmptex 6436
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 6342.) 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 3683 . . . 4 (𝑦 = 𝐶𝑦 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
2 fvmptex.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
3 nfcv 2912 . . . . . 6 𝑦𝐵
4 nfcsb1v 3696 . . . . . 6 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3689 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 4881 . . . . 5 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
72, 6eqtri 2792 . . . 4 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
81, 7fvmpti 6423 . . 3 (𝐶𝐴 → (𝐹𝐶) = ( I ‘𝐶 / 𝑥𝐵))
91fveq2d 6336 . . . 4 (𝑦 = 𝐶 → ( I ‘𝑦 / 𝑥𝐵) = ( I ‘𝐶 / 𝑥𝐵))
10 fvmptex.2 . . . . 5 𝐺 = (𝑥𝐴 ↦ ( I ‘𝐵))
11 nfcv 2912 . . . . . 6 𝑦( I ‘𝐵)
12 nfcv 2912 . . . . . . 7 𝑥 I
1312, 4nffv 6339 . . . . . 6 𝑥( I ‘𝑦 / 𝑥𝐵)
145fveq2d 6336 . . . . . 6 (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘𝑦 / 𝑥𝐵))
1511, 13, 14cbvmpt 4881 . . . . 5 (𝑥𝐴 ↦ ( I ‘𝐵)) = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
1610, 15eqtri 2792 . . . 4 𝐺 = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
17 fvex 6342 . . . 4 ( I ‘𝐶 / 𝑥𝐵) ∈ V
189, 16, 17fvmpt 6424 . . 3 (𝐶𝐴 → (𝐺𝐶) = ( I ‘𝐶 / 𝑥𝐵))
198, 18eqtr4d 2807 . 2 (𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
202dmmptss 5775 . . . . . 6 dom 𝐹𝐴
2120sseli 3746 . . . . 5 (𝐶 ∈ dom 𝐹𝐶𝐴)
2221con3i 151 . . . 4 𝐶𝐴 → ¬ 𝐶 ∈ dom 𝐹)
23 ndmfv 6359 . . . 4 𝐶 ∈ dom 𝐹 → (𝐹𝐶) = ∅)
2422, 23syl 17 . . 3 𝐶𝐴 → (𝐹𝐶) = ∅)
25 fvex 6342 . . . . . 6 ( I ‘𝐵) ∈ V
2625, 10dmmpti 6163 . . . . 5 dom 𝐺 = 𝐴
2726eleq2i 2841 . . . 4 (𝐶 ∈ dom 𝐺𝐶𝐴)
28 ndmfv 6359 . . . 4 𝐶 ∈ dom 𝐺 → (𝐺𝐶) = ∅)
2927, 28sylnbir 320 . . 3 𝐶𝐴 → (𝐺𝐶) = ∅)
3024, 29eqtr4d 2807 . 2 𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
3119, 30pm2.61i 176 1 (𝐹𝐶) = (𝐺𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1630  wcel 2144  csb 3680  c0 4061  cmpt 4861   I cid 5156  dom cdm 5249  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  fvmptnf  6444  sumeq2ii  14630  prodeq2ii  14849
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