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Theorem fvmptss2 6900
Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypotheses
Ref Expression
fvmptn.1 (𝑥 = 𝐷𝐵 = 𝐶)
fvmptn.2 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmptss2 (𝐹𝐷) ⊆ 𝐶
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmptss2
StepHypRef Expression
1 fvmptn.1 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐶)
21eleq1d 2823 . . . 4 (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
3 fvmptn.2 . . . . 5 𝐹 = (𝑥𝐴𝐵)
43dmmpt 6143 . . . 4 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
52, 4elrab2 3627 . . 3 (𝐷 ∈ dom 𝐹 ↔ (𝐷𝐴𝐶 ∈ V))
61, 3fvmptg 6873 . . . 4 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) = 𝐶)
7 eqimss 3977 . . . 4 ((𝐹𝐷) = 𝐶 → (𝐹𝐷) ⊆ 𝐶)
86, 7syl 17 . . 3 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) ⊆ 𝐶)
95, 8sylbi 216 . 2 (𝐷 ∈ dom 𝐹 → (𝐹𝐷) ⊆ 𝐶)
10 ndmfv 6804 . . 3 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
11 0ss 4330 . . 3 ∅ ⊆ 𝐶
1210, 11eqsstrdi 3975 . 2 𝐷 ∈ dom 𝐹 → (𝐹𝐷) ⊆ 𝐶)
139, 12pm2.61i 182 1 (𝐹𝐷) ⊆ 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  wss 3887  c0 4256  cmpt 5157  dom cdm 5589  cfv 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441
This theorem is referenced by:  cvmsi  33227
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