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Theorem fvmptex 6832
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 6730.) 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 3814 . . . 4 (𝑦 = 𝐶𝑦 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
2 fvmptex.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
3 nfcv 2904 . . . . . 6 𝑦𝐵
4 nfcsb1v 3836 . . . . . 6 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3825 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 5156 . . . . 5 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
72, 6eqtri 2765 . . . 4 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
81, 7fvmpti 6817 . . 3 (𝐶𝐴 → (𝐹𝐶) = ( I ‘𝐶 / 𝑥𝐵))
91fveq2d 6721 . . . 4 (𝑦 = 𝐶 → ( I ‘𝑦 / 𝑥𝐵) = ( I ‘𝐶 / 𝑥𝐵))
10 fvmptex.2 . . . . 5 𝐺 = (𝑥𝐴 ↦ ( I ‘𝐵))
11 nfcv 2904 . . . . . 6 𝑦( I ‘𝐵)
12 nfcv 2904 . . . . . . 7 𝑥 I
1312, 4nffv 6727 . . . . . 6 𝑥( I ‘𝑦 / 𝑥𝐵)
145fveq2d 6721 . . . . . 6 (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘𝑦 / 𝑥𝐵))
1511, 13, 14cbvmpt 5156 . . . . 5 (𝑥𝐴 ↦ ( I ‘𝐵)) = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
1610, 15eqtri 2765 . . . 4 𝐺 = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
17 fvex 6730 . . . 4 ( I ‘𝐶 / 𝑥𝐵) ∈ V
189, 16, 17fvmpt 6818 . . 3 (𝐶𝐴 → (𝐺𝐶) = ( I ‘𝐶 / 𝑥𝐵))
198, 18eqtr4d 2780 . 2 (𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
202dmmptss 6104 . . . . 5 dom 𝐹𝐴
2120sseli 3896 . . . 4 (𝐶 ∈ dom 𝐹𝐶𝐴)
22 ndmfv 6747 . . . 4 𝐶 ∈ dom 𝐹 → (𝐹𝐶) = ∅)
2321, 22nsyl5 162 . . 3 𝐶𝐴 → (𝐹𝐶) = ∅)
24 fvex 6730 . . . . . 6 ( I ‘𝐵) ∈ V
2524, 10dmmpti 6522 . . . . 5 dom 𝐺 = 𝐴
2625eleq2i 2829 . . . 4 (𝐶 ∈ dom 𝐺𝐶𝐴)
27 ndmfv 6747 . . . 4 𝐶 ∈ dom 𝐺 → (𝐺𝐶) = ∅)
2826, 27sylnbir 334 . . 3 𝐶𝐴 → (𝐺𝐶) = ∅)
2923, 28eqtr4d 2780 . 2 𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
3019, 29pm2.61i 185 1 (𝐹𝐶) = (𝐺𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1543  wcel 2110  csb 3811  c0 4237  cmpt 5135   I cid 5454  dom cdm 5551  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388
This theorem is referenced by:  fvmptnf  6840  sumeq2ii  15257  prodeq2ii  15475
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