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Theorem fvmptssdm 5505
 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 5421 . . . . . 6 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
21sseq1d 3126 . . . . 5 (𝑦 = 𝐷 → ((𝐹𝑦) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
32imbi2d 229 . . . 4 (𝑦 = 𝐷 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)))
4 nfrab1 2610 . . . . . . 7 𝑥{𝑥𝐴𝐵 ∈ V}
54nfcri 2275 . . . . . 6 𝑥 𝑦 ∈ {𝑥𝐴𝐵 ∈ V}
6 nfra1 2466 . . . . . . 7 𝑥𝑥𝐴 𝐵𝐶
7 fvmpt2.1 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
8 nfmpt1 4021 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
97, 8nfcxfr 2278 . . . . . . . . 9 𝑥𝐹
10 nfcv 2281 . . . . . . . . 9 𝑥𝑦
119, 10nffv 5431 . . . . . . . 8 𝑥(𝐹𝑦)
12 nfcv 2281 . . . . . . . 8 𝑥𝐶
1311, 12nfss 3090 . . . . . . 7 𝑥(𝐹𝑦) ⊆ 𝐶
146, 13nfim 1551 . . . . . 6 𝑥(∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)
155, 14nfim 1551 . . . . 5 𝑥(𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
16 eleq1 2202 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} ↔ 𝑦 ∈ {𝑥𝐴𝐵 ∈ V}))
17 fveq2 5421 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1817sseq1d 3126 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ⊆ 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
1918imbi2d 229 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)))
2016, 19imbi12d 233 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶)) ↔ (𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))))
217dmmpt 5034 . . . . . . 7 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
2221eleq2i 2206 . . . . . 6 (𝑥 ∈ dom 𝐹𝑥 ∈ {𝑥𝐴𝐵 ∈ V})
2321rabeq2i 2683 . . . . . . . . . 10 (𝑥 ∈ dom 𝐹 ↔ (𝑥𝐴𝐵 ∈ V))
247fvmpt2 5504 . . . . . . . . . . 11 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
25 eqimss 3151 . . . . . . . . . . 11 ((𝐹𝑥) = 𝐵 → (𝐹𝑥) ⊆ 𝐵)
2624, 25syl 14 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) ⊆ 𝐵)
2723, 26sylbi 120 . . . . . . . . 9 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
2827adantr 274 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐵)
297dmmptss 5035 . . . . . . . . . 10 dom 𝐹𝐴
3029sseli 3093 . . . . . . . . 9 (𝑥 ∈ dom 𝐹𝑥𝐴)
31 rsp 2480 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
3230, 31mpan9 279 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝐵𝐶)
3328, 32sstrd 3107 . . . . . . 7 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐶)
3433ex 114 . . . . . 6 (𝑥 ∈ dom 𝐹 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
3522, 34sylbir 134 . . . . 5 (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
3615, 20, 35chvar 1730 . . . 4 (𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
373, 36vtoclga 2752 . . 3 (𝐷 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3837, 21eleq2s 2234 . 2 (𝐷 ∈ dom 𝐹 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3938imp 123 1 ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝐷) ⊆ 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2416  {crab 2420  Vcvv 2686   ⊆ wss 3071   ↦ cmpt 3989  dom cdm 4539  ‘cfv 5123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fv 5131 This theorem is referenced by: (None)
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