MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptss2 Structured version   Visualization version   GIF version

Theorem fvmptss2 6446
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 2834 . . . 4 (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
3 fvmptn.2 . . . . 5 𝐹 = (𝑥𝐴𝐵)
43dmmpt 5774 . . . 4 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
52, 4elrab2 3516 . . 3 (𝐷 ∈ dom 𝐹 ↔ (𝐷𝐴𝐶 ∈ V))
61, 3fvmptg 6422 . . . 4 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) = 𝐶)
7 eqimss 3804 . . . 4 ((𝐹𝐷) = 𝐶 → (𝐹𝐷) ⊆ 𝐶)
86, 7syl 17 . . 3 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) ⊆ 𝐶)
95, 8sylbi 207 . 2 (𝐷 ∈ dom 𝐹 → (𝐹𝐷) ⊆ 𝐶)
10 ndmfv 6359 . . 3 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
11 0ss 4114 . . 3 ∅ ⊆ 𝐶
1210, 11syl6eqss 3802 . 2 𝐷 ∈ dom 𝐹 → (𝐹𝐷) ⊆ 𝐶)
139, 12pm2.61i 176 1 (𝐹𝐷) ⊆ 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  wss 3721  c0 4061  cmpt 4861  dom cdm 5249  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039
This theorem is referenced by:  cvmsi  31579
  Copyright terms: Public domain W3C validator