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Theorem fvmptss2 6260
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 2683 . . . 4 (𝑥 = 𝐷 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
3 fvmptn.2 . . . . 5 𝐹 = (𝑥𝐴𝐵)
43dmmpt 5589 . . . 4 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
52, 4elrab2 3348 . . 3 (𝐷 ∈ dom 𝐹 ↔ (𝐷𝐴𝐶 ∈ V))
61, 3fvmptg 6237 . . . 4 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) = 𝐶)
7 eqimss 3636 . . . 4 ((𝐹𝐷) = 𝐶 → (𝐹𝐷) ⊆ 𝐶)
86, 7syl 17 . . 3 ((𝐷𝐴𝐶 ∈ V) → (𝐹𝐷) ⊆ 𝐶)
95, 8sylbi 207 . 2 (𝐷 ∈ dom 𝐹 → (𝐹𝐷) ⊆ 𝐶)
10 ndmfv 6175 . . 3 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
11 0ss 3944 . . 3 ∅ ⊆ 𝐶
1210, 11syl6eqss 3634 . 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 384   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555  c0 3891  cmpt 4673  dom cdm 5074  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855
This theorem is referenced by:  cvmsi  30955
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