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

Theorem fvmptss2 6800
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 2817 . . . 4 (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
3 fvmptn.2 . . . . 5 𝐹 = (𝑥𝐴𝐵)
43dmmpt 6072 . . . 4 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
52, 4elrab2 3591 . . 3 (𝐷 ∈ dom 𝐹 ↔ (𝐷𝐴𝐶 ∈ V))
61, 3fvmptg 6773 . . . 4 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) = 𝐶)
7 eqimss 3933 . . . 4 ((𝐹𝐷) = 𝐶 → (𝐹𝐷) ⊆ 𝐶)
86, 7syl 17 . . 3 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) ⊆ 𝐶)
95, 8sylbi 220 . 2 (𝐷 ∈ dom 𝐹 → (𝐹𝐷) ⊆ 𝐶)
10 ndmfv 6704 . . 3 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
11 0ss 4285 . . 3 ∅ ⊆ 𝐶
1210, 11eqsstrdi 3931 . 2 𝐷 ∈ dom 𝐹 → (𝐹𝐷) ⊆ 𝐶)
139, 12pm2.61i 185 1 (𝐹𝐷) ⊆ 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2114  Vcvv 3398  wss 3843  c0 4211  cmpt 5110  dom cdm 5525  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fv 6347
This theorem is referenced by:  cvmsi  32798
  Copyright terms: Public domain W3C validator