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Theorem fvmptssdm 5180
 Description: If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)
Hypothesis
Ref Expression
fvmpt2.1 𝐹 = (x AB)
Assertion
Ref Expression
fvmptssdm ((𝐷 dom 𝐹 x A B𝐶) → (𝐹𝐷) ⊆ 𝐶)
Distinct variable groups:   x,A   x,𝐶
Allowed substitution hints:   B(x)   𝐷(x)   𝐹(x)

Proof of Theorem fvmptssdm
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 fveq2 5103 . . . . . 6 (y = 𝐷 → (𝐹y) = (𝐹𝐷))
21sseq1d 2949 . . . . 5 (y = 𝐷 → ((𝐹y) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
32imbi2d 219 . . . 4 (y = 𝐷 → ((x A B𝐶 → (𝐹y) ⊆ 𝐶) ↔ (x A B𝐶 → (𝐹𝐷) ⊆ 𝐶)))
4 nfrab1 2467 . . . . . . 7 x{x AB V}
54nfcri 2154 . . . . . 6 x y {x AB V}
6 nfra1 2333 . . . . . . 7 xx A B𝐶
7 fvmpt2.1 . . . . . . . . . 10 𝐹 = (x AB)
8 nfmpt1 3824 . . . . . . . . . 10 x(x AB)
97, 8nfcxfr 2157 . . . . . . . . 9 x𝐹
10 nfcv 2160 . . . . . . . . 9 xy
119, 10nffv 5110 . . . . . . . 8 x(𝐹y)
12 nfcv 2160 . . . . . . . 8 x𝐶
1311, 12nfss 2915 . . . . . . 7 x(𝐹y) ⊆ 𝐶
146, 13nfim 1446 . . . . . 6 x(x A B𝐶 → (𝐹y) ⊆ 𝐶)
155, 14nfim 1446 . . . . 5 x(y {x AB V} → (x A B𝐶 → (𝐹y) ⊆ 𝐶))
16 eleq1 2082 . . . . . 6 (x = y → (x {x AB V} ↔ y {x AB V}))
17 fveq2 5103 . . . . . . . 8 (x = y → (𝐹x) = (𝐹y))
1817sseq1d 2949 . . . . . . 7 (x = y → ((𝐹x) ⊆ 𝐶 ↔ (𝐹y) ⊆ 𝐶))
1918imbi2d 219 . . . . . 6 (x = y → ((x A B𝐶 → (𝐹x) ⊆ 𝐶) ↔ (x A B𝐶 → (𝐹y) ⊆ 𝐶)))
2016, 19imbi12d 223 . . . . 5 (x = y → ((x {x AB V} → (x A B𝐶 → (𝐹x) ⊆ 𝐶)) ↔ (y {x AB V} → (x A B𝐶 → (𝐹y) ⊆ 𝐶))))
217dmmpt 4743 . . . . . . 7 dom 𝐹 = {x AB V}
2221eleq2i 2086 . . . . . 6 (x dom 𝐹x {x AB V})
2321rabeq2i 2532 . . . . . . . . . 10 (x dom 𝐹 ↔ (x A B V))
247fvmpt2 5179 . . . . . . . . . . 11 ((x A B V) → (𝐹x) = B)
25 eqimss 2974 . . . . . . . . . . 11 ((𝐹x) = B → (𝐹x) ⊆ B)
2624, 25syl 14 . . . . . . . . . 10 ((x A B V) → (𝐹x) ⊆ B)
2723, 26sylbi 114 . . . . . . . . 9 (x dom 𝐹 → (𝐹x) ⊆ B)
2827adantr 261 . . . . . . . 8 ((x dom 𝐹 x A B𝐶) → (𝐹x) ⊆ B)
297dmmptss 4744 . . . . . . . . . 10 dom 𝐹A
3029sseli 2918 . . . . . . . . 9 (x dom 𝐹x A)
31 rsp 2347 . . . . . . . . 9 (x A B𝐶 → (x AB𝐶))
3230, 31mpan9 265 . . . . . . . 8 ((x dom 𝐹 x A B𝐶) → B𝐶)
3328, 32sstrd 2932 . . . . . . 7 ((x dom 𝐹 x A B𝐶) → (𝐹x) ⊆ 𝐶)
3433ex 108 . . . . . 6 (x dom 𝐹 → (x A B𝐶 → (𝐹x) ⊆ 𝐶))
3522, 34sylbir 125 . . . . 5 (x {x AB V} → (x A B𝐶 → (𝐹x) ⊆ 𝐶))
3615, 20, 35chvar 1622 . . . 4 (y {x AB V} → (x A B𝐶 → (𝐹y) ⊆ 𝐶))
373, 36vtoclga 2596 . . 3 (𝐷 {x AB V} → (x A B𝐶 → (𝐹𝐷) ⊆ 𝐶))
3837, 21eleq2s 2114 . 2 (𝐷 dom 𝐹 → (x A B𝐶 → (𝐹𝐷) ⊆ 𝐶))
3938imp 115 1 ((𝐷 dom 𝐹 x A B𝐶) → (𝐹𝐷) ⊆ 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  ∀wral 2284  {crab 2288  Vcvv 2535   ⊆ wss 2894   ↦ cmpt 3792  dom cdm 4272  ‘cfv 4829 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fv 4837 This theorem is referenced by: (None)
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