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Theorem fvmptssdm 5198
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 5121 . . . . . 6 (y = 𝐷 → (𝐹y) = (𝐹𝐷))
21sseq1d 2966 . . . . 5 (y = 𝐷 → ((𝐹y) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
32imbi2d 219 . . . 4 (y = 𝐷 → ((x A B𝐶 → (𝐹y) ⊆ 𝐶) ↔ (x A B𝐶 → (𝐹𝐷) ⊆ 𝐶)))
4 nfrab1 2483 . . . . . . 7 x{x AB V}
54nfcri 2169 . . . . . 6 x y {x AB V}
6 nfra1 2349 . . . . . . 7 xx A B𝐶
7 fvmpt2.1 . . . . . . . . . 10 𝐹 = (x AB)
8 nfmpt1 3841 . . . . . . . . . 10 x(x AB)
97, 8nfcxfr 2172 . . . . . . . . 9 x𝐹
10 nfcv 2175 . . . . . . . . 9 xy
119, 10nffv 5128 . . . . . . . 8 x(𝐹y)
12 nfcv 2175 . . . . . . . 8 x𝐶
1311, 12nfss 2932 . . . . . . 7 x(𝐹y) ⊆ 𝐶
146, 13nfim 1461 . . . . . 6 x(x A B𝐶 → (𝐹y) ⊆ 𝐶)
155, 14nfim 1461 . . . . 5 x(y {x AB V} → (x A B𝐶 → (𝐹y) ⊆ 𝐶))
16 eleq1 2097 . . . . . 6 (x = y → (x {x AB V} ↔ y {x AB V}))
17 fveq2 5121 . . . . . . . 8 (x = y → (𝐹x) = (𝐹y))
1817sseq1d 2966 . . . . . . 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 4759 . . . . . . 7 dom 𝐹 = {x AB V}
2221eleq2i 2101 . . . . . 6 (x dom 𝐹x {x AB V})
2321rabeq2i 2548 . . . . . . . . . 10 (x dom 𝐹 ↔ (x A B V))
247fvmpt2 5197 . . . . . . . . . . 11 ((x A B V) → (𝐹x) = B)
25 eqimss 2991 . . . . . . . . . . 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 4760 . . . . . . . . . 10 dom 𝐹A
3029sseli 2935 . . . . . . . . 9 (x dom 𝐹x A)
31 rsp 2363 . . . . . . . . 9 (x A B𝐶 → (x AB𝐶))
3230, 31mpan9 265 . . . . . . . 8 ((x dom 𝐹 x A B𝐶) → B𝐶)
3328, 32sstrd 2949 . . . . . . 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 1637 . . . 4 (y {x AB V} → (x A B𝐶 → (𝐹y) ⊆ 𝐶))
373, 36vtoclga 2613 . . 3 (𝐷 {x AB V} → (x A B𝐶 → (𝐹𝐷) ⊆ 𝐶))
3837, 21eleq2s 2129 . 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 1242   wcel 1390  wral 2300  {crab 2304  Vcvv 2551  wss 2911  cmpt 3809  dom cdm 4288  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by: (None)
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