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Theorem fvmptssdm 5282
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 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmptssdm ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥𝐴 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 5205 . . . . . 6 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
21sseq1d 2999 . . . . 5 (𝑦 = 𝐷 → ((𝐹𝑦) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
32imbi2d 223 . . . 4 (𝑦 = 𝐷 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)))
4 nfrab1 2506 . . . . . . 7 𝑥{𝑥𝐴𝐵 ∈ V}
54nfcri 2188 . . . . . 6 𝑥 𝑦 ∈ {𝑥𝐴𝐵 ∈ V}
6 nfra1 2372 . . . . . . 7 𝑥𝑥𝐴 𝐵𝐶
7 fvmpt2.1 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
8 nfmpt1 3877 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
97, 8nfcxfr 2191 . . . . . . . . 9 𝑥𝐹
10 nfcv 2194 . . . . . . . . 9 𝑥𝑦
119, 10nffv 5212 . . . . . . . 8 𝑥(𝐹𝑦)
12 nfcv 2194 . . . . . . . 8 𝑥𝐶
1311, 12nfss 2965 . . . . . . 7 𝑥(𝐹𝑦) ⊆ 𝐶
146, 13nfim 1480 . . . . . 6 𝑥(∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)
155, 14nfim 1480 . . . . 5 𝑥(𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
16 eleq1 2116 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} ↔ 𝑦 ∈ {𝑥𝐴𝐵 ∈ V}))
17 fveq2 5205 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1817sseq1d 2999 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ⊆ 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
1918imbi2d 223 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)))
2016, 19imbi12d 227 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶)) ↔ (𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))))
217dmmpt 4843 . . . . . . 7 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
2221eleq2i 2120 . . . . . 6 (𝑥 ∈ dom 𝐹𝑥 ∈ {𝑥𝐴𝐵 ∈ V})
2321rabeq2i 2571 . . . . . . . . . 10 (𝑥 ∈ dom 𝐹 ↔ (𝑥𝐴𝐵 ∈ V))
247fvmpt2 5281 . . . . . . . . . . 11 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
25 eqimss 3024 . . . . . . . . . . 11 ((𝐹𝑥) = 𝐵 → (𝐹𝑥) ⊆ 𝐵)
2624, 25syl 14 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) ⊆ 𝐵)
2723, 26sylbi 118 . . . . . . . . 9 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
2827adantr 265 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐵)
297dmmptss 4844 . . . . . . . . . 10 dom 𝐹𝐴
3029sseli 2968 . . . . . . . . 9 (𝑥 ∈ dom 𝐹𝑥𝐴)
31 rsp 2386 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
3230, 31mpan9 269 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝐵𝐶)
3328, 32sstrd 2982 . . . . . . 7 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐶)
3433ex 112 . . . . . 6 (𝑥 ∈ dom 𝐹 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
3522, 34sylbir 129 . . . . 5 (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
3615, 20, 35chvar 1656 . . . 4 (𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
373, 36vtoclga 2636 . . 3 (𝐷 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3837, 21eleq2s 2148 . 2 (𝐷 ∈ dom 𝐹 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3938imp 119 1 ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝐷) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  wral 2323  {crab 2327  Vcvv 2574  wss 2944  cmpt 3845  dom cdm 4372  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fv 4937
This theorem is referenced by: (None)
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