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Theorem fvmptss2 6977
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 2819 . . . 4 (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
3 fvmptn.2 . . . . 5 𝐹 = (𝑥𝐴𝐵)
43dmmpt 6196 . . . 4 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
52, 4elrab2 3652 . . 3 (𝐷 ∈ dom 𝐹 ↔ (𝐷𝐴𝐶 ∈ V))
61, 3fvmptg 6950 . . . 4 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) = 𝐶)
7 eqimss 4004 . . . 4 ((𝐹𝐷) = 𝐶 → (𝐹𝐷) ⊆ 𝐶)
86, 7syl 17 . . 3 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) ⊆ 𝐶)
95, 8sylbi 216 . 2 (𝐷 ∈ dom 𝐹 → (𝐹𝐷) ⊆ 𝐶)
10 ndmfv 6881 . . 3 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
11 0ss 4360 . . 3 ∅ ⊆ 𝐶
1210, 11eqsstrdi 4002 . 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 397   = wceq 1542  wcel 2107  Vcvv 3447  wss 3914  c0 4286  cmpt 5192  dom cdm 5637  cfv 6500
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fv 6508
This theorem is referenced by:  cvmsi  33923
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