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Theorem fvmptex 7001
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 6894.) 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 3894 . . . 4 (𝑦 = 𝐶𝑦 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
2 fvmptex.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
3 nfcv 2904 . . . . . 6 𝑦𝐵
4 nfcsb1v 3916 . . . . . 6 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3905 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 5255 . . . . 5 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
72, 6eqtri 2761 . . . 4 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
81, 7fvmpti 6986 . . 3 (𝐶𝐴 → (𝐹𝐶) = ( I ‘𝐶 / 𝑥𝐵))
91fveq2d 6885 . . . 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 6891 . . . . . 6 𝑥( I ‘𝑦 / 𝑥𝐵)
145fveq2d 6885 . . . . . 6 (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘𝑦 / 𝑥𝐵))
1511, 13, 14cbvmpt 5255 . . . . 5 (𝑥𝐴 ↦ ( I ‘𝐵)) = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
1610, 15eqtri 2761 . . . 4 𝐺 = (𝑦𝐴 ↦ ( I ‘𝑦 / 𝑥𝐵))
17 fvex 6894 . . . 4 ( I ‘𝐶 / 𝑥𝐵) ∈ V
189, 16, 17fvmpt 6987 . . 3 (𝐶𝐴 → (𝐺𝐶) = ( I ‘𝐶 / 𝑥𝐵))
198, 18eqtr4d 2776 . 2 (𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
202dmmptss 6232 . . . . 5 dom 𝐹𝐴
2120sseli 3976 . . . 4 (𝐶 ∈ dom 𝐹𝐶𝐴)
22 ndmfv 6916 . . . 4 𝐶 ∈ dom 𝐹 → (𝐹𝐶) = ∅)
2321, 22nsyl5 159 . . 3 𝐶𝐴 → (𝐹𝐶) = ∅)
24 fvex 6894 . . . . . 6 ( I ‘𝐵) ∈ V
2524, 10dmmpti 6684 . . . . 5 dom 𝐺 = 𝐴
2625eleq2i 2826 . . . 4 (𝐶 ∈ dom 𝐺𝐶𝐴)
27 ndmfv 6916 . . . 4 𝐶 ∈ dom 𝐺 → (𝐺𝐶) = ∅)
2826, 27sylnbir 331 . . 3 𝐶𝐴 → (𝐺𝐶) = ∅)
2923, 28eqtr4d 2776 . 2 𝐶𝐴 → (𝐹𝐶) = (𝐺𝐶))
3019, 29pm2.61i 182 1 (𝐹𝐶) = (𝐺𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  csb 3891  c0 4320  cmpt 5227   I cid 5569  dom cdm 5672  cfv 6535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-fv 6543
This theorem is referenced by:  fvmptnf  7009  sumeq2ii  15626  prodeq2ii  15844
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