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Theorem fvmptss 5705
 Description: If all the values of the mapping are subsets of a class C, then so is any evaluation of the mapping, even if D is not in the base set A. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypothesis
Ref Expression
fvmpt2.1 F = (x A B)
Assertion
Ref Expression
fvmptss (x A B C → (FD) C)
Distinct variable groups:   x,A   x,C
Allowed substitution hints:   B(x)   D(x)   F(x)

Proof of Theorem fvmptss
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 fvmpt2.1 . . . . 5 F = (x A B)
21dmmptss 5685 . . . 4 dom F A
32sseli 3269 . . 3 (D dom FD A)
4 fveq2 5328 . . . . . . 7 (y = D → (Fy) = (FD))
54sseq1d 3298 . . . . . 6 (y = D → ((Fy) C ↔ (FD) C))
65imbi2d 307 . . . . 5 (y = D → ((x A B C → (Fy) C) ↔ (x A B C → (FD) C)))
7 nfcv 2489 . . . . . 6 xy
8 nfra1 2664 . . . . . . 7 xx A B C
9 nfmpt1 5672 . . . . . . . . . 10 x(x A B)
101, 9nfcxfr 2486 . . . . . . . . 9 xF
1110, 7nffv 5334 . . . . . . . 8 x(Fy)
12 nfcv 2489 . . . . . . . 8 xC
1311, 12nfss 3266 . . . . . . 7 x(Fy) C
148, 13nfim 1813 . . . . . 6 x(x A B C → (Fy) C)
15 fveq2 5328 . . . . . . . 8 (x = y → (Fx) = (Fy))
1615sseq1d 3298 . . . . . . 7 (x = y → ((Fx) C ↔ (Fy) C))
1716imbi2d 307 . . . . . 6 (x = y → ((x A B C → (Fx) C) ↔ (x A B C → (Fy) C)))
181dmmpt 5683 . . . . . . . . . . 11 dom F = {x A B V}
1918rabeq2i 2856 . . . . . . . . . 10 (x dom F ↔ (x A B V))
201fvmpt2 5704 . . . . . . . . . . 11 ((x A B V) → (Fx) = B)
21 eqimss 3323 . . . . . . . . . . 11 ((Fx) = B → (Fx) B)
2220, 21syl 15 . . . . . . . . . 10 ((x A B V) → (Fx) B)
2319, 22sylbi 187 . . . . . . . . 9 (x dom F → (Fx) B)
24 ndmfv 5349 . . . . . . . . . 10 x dom F → (Fx) = )
25 0ss 3579 . . . . . . . . . . 11 B
2625a1i 10 . . . . . . . . . 10 x dom F B)
2724, 26eqsstrd 3305 . . . . . . . . 9 x dom F → (Fx) B)
2823, 27pm2.61i 156 . . . . . . . 8 (Fx) B
29 rsp 2674 . . . . . . . . 9 (x A B C → (x AB C))
3029impcom 419 . . . . . . . 8 ((x A x A B C) → B C)
3128, 30syl5ss 3283 . . . . . . 7 ((x A x A B C) → (Fx) C)
3231ex 423 . . . . . 6 (x A → (x A B C → (Fx) C))
337, 14, 17, 32vtoclgaf 2919 . . . . 5 (y A → (x A B C → (Fy) C))
346, 33vtoclga 2920 . . . 4 (D A → (x A B C → (FD) C))
3534impcom 419 . . 3 ((x A B C D A) → (FD) C)
363, 35sylan2 460 . 2 ((x A B C D dom F) → (FD) C)
37 ndmfv 5349 . . . 4 D dom F → (FD) = )
3837adantl 452 . . 3 ((x A B C ¬ D dom F) → (FD) = )
39 0ss 3579 . . . 4 C
4039a1i 10 . . 3 ((x A B C ¬ D dom F) → C)
4138, 40eqsstrd 3305 . 2 ((x A B C ¬ D dom F) → (FD) C)
4236, 41pm2.61dan 766 1 (x A B C → (FD) C)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ⊆ wss 3257  ∅c0 3550  dom cdm 4772   ‘cfv 4781   ↦ cmpt 5651 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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: (None)
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