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Theorem fvmptex 6759
 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 6658.) 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 3831 . . . 4 (𝑦 = 𝐶𝑦 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
2 fvmptex.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
3 nfcv 2955 . . . . . 6 𝑦𝐵
4 nfcsb1v 3852 . . . . . 6 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3842 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 5131 . . . . 5 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
72, 6eqtri 2821 . . . 4 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
81, 7fvmpti 6744 . . 3 (𝐶𝐴 → (𝐹𝐶) = ( I ‘𝐶 / 𝑥𝐵))
91fveq2d 6649 . . . 4 (𝑦 = 𝐶 → ( I ‘𝑦 / 𝑥𝐵) = ( I ‘𝐶 / 𝑥𝐵))
10 fvmptex.2 . . . . 5 𝐺 = (𝑥𝐴 ↦ ( I ‘𝐵))
11 nfcv 2955 . . . . . 6 𝑦( I ‘𝐵)
12 nfcv 2955 . . . . . . 7 𝑥 I
1312, 4nffv 6655 . . . . . 6 𝑥( I ‘𝑦 / 𝑥𝐵)
145fveq2d 6649 . . . . . 6 (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘𝑦 / 𝑥𝐵))
1511, 13, 14cbvmpt 5131 . . . . 5 (𝑥𝐴 ↦ ( I ‘𝐵)) = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
1610, 15eqtri 2821 . . . 4 𝐺 = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
17 fvex 6658 . . . 4 ( I ‘𝐶 / 𝑥𝐵) ∈ V
189, 16, 17fvmpt 6745 . . 3 (𝐶𝐴 → (𝐺𝐶) = ( I ‘𝐶 / 𝑥𝐵))
198, 18eqtr4d 2836 . 2 (𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
202dmmptss 6062 . . . . 5 dom 𝐹𝐴
2120sseli 3911 . . . 4 (𝐶 ∈ dom 𝐹𝐶𝐴)
22 ndmfv 6675 . . . 4 𝐶 ∈ dom 𝐹 → (𝐹𝐶) = ∅)
2321, 22nsyl5 162 . . 3 𝐶𝐴 → (𝐹𝐶) = ∅)
24 fvex 6658 . . . . . 6 ( I ‘𝐵) ∈ V
2524, 10dmmpti 6464 . . . . 5 dom 𝐺 = 𝐴
2625eleq2i 2881 . . . 4 (𝐶 ∈ dom 𝐺𝐶𝐴)
27 ndmfv 6675 . . . 4 𝐶 ∈ dom 𝐺 → (𝐺𝐶) = ∅)
2826, 27sylnbir 334 . . 3 𝐶𝐴 → (𝐺𝐶) = ∅)
2923, 28eqtr4d 2836 . 2 𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
3019, 29pm2.61i 185 1 (𝐹𝐶) = (𝐺𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  ⦋csb 3828  ∅c0 4243   ↦ cmpt 5110   I cid 5424  dom cdm 5519  ‘cfv 6324 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332 This theorem is referenced by:  fvmptnf  6767  sumeq2ii  15044  prodeq2ii  15261
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