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Theorem fvmptex 5721
 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 5339.) 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 = (x A B)
fvmptex.2 G = (x A ( 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 3139 . . . 4 (y = C[y / x]B = [C / x]B)
2 fvmptex.1 . . . . 5 F = (x A B)
3 nfcv 2489 . . . . . 6 yB
4 nfcsb1v 3168 . . . . . 6 x[y / x]B
5 csbeq1a 3144 . . . . . 6 (x = yB = [y / x]B)
63, 4, 5cbvmpt 5676 . . . . 5 (x A B) = (y A [y / x]B)
72, 6eqtri 2373 . . . 4 F = (y A [y / x]B)
81, 7fvmpti 5699 . . 3 (C A → (FC) = ( I ‘[C / x]B))
91fveq2d 5332 . . . 4 (y = C → ( I ‘[y / x]B) = ( I ‘[C / x]B))
10 fvmptex.2 . . . . 5 G = (x A ( I ‘B))
11 nfcv 2489 . . . . . 6 y( I ‘B)
12 nfcv 2489 . . . . . . 7 x I
1312, 4nffv 5334 . . . . . 6 x( I ‘[y / x]B)
145fveq2d 5332 . . . . . 6 (x = y → ( I ‘B) = ( I ‘[y / x]B))
1511, 13, 14cbvmpt 5676 . . . . 5 (x A ( I ‘B)) = (y A ( I ‘[y / x]B))
1610, 15eqtri 2373 . . . 4 G = (y A ( I ‘[y / x]B))
17 fvex 5339 . . . 4 ( I ‘[C / x]B) V
189, 16, 17fvmpt 5700 . . 3 (C A → (GC) = ( I ‘[C / x]B))
198, 18eqtr4d 2388 . 2 (C A → (FC) = (GC))
202dmmptss 5685 . . . . . 6 dom F A
2120sseli 3269 . . . . 5 (C dom FC A)
2221con3i 127 . . . 4 C A → ¬ C dom F)
23 ndmfv 5349 . . . 4 C dom F → (FC) = )
2422, 23syl 15 . . 3 C A → (FC) = )
25 fvex 5339 . . . . . 6 ( I ‘B) V
2625, 10dmmpti 5691 . . . . 5 dom G = A
2726eleq2i 2417 . . . 4 (C dom GC A)
28 ndmfv 5349 . . . 4 C dom G → (GC) = )
2927, 28sylnbir 298 . . 3 C A → (GC) = )
3024, 29eqtr4d 2388 . 2 C A → (FC) = (GC))
3119, 30pm2.61i 156 1 (FC) = (GC)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  [csb 3136  ∅c0 3550   I cid 4763  dom cdm 4772   ‘cfv 4781   ↦ cmpt 5651 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-mpt 5652 This theorem is referenced by:  fvmptnf  5723
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