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Theorem fvmptss2 5172
 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.) (Revised by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
fvmptss2.1 (x = 𝐷B = 𝐶)
fvmptss2.2 𝐹 = (x AB)
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
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 fvss 5114 . 2 (y(𝐷𝐹yy𝐶) → (𝐹𝐷) ⊆ 𝐶)
2 fvmptss2.2 . . . . . 6 𝐹 = (x AB)
32funmpt2 4865 . . . . 5 Fun 𝐹
4 funrel 4845 . . . . 5 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 7 . . . 4 Rel 𝐹
65brrelexi 4311 . . 3 (𝐷𝐹y𝐷 V)
7 nfcv 2160 . . . 4 x𝐷
8 nfmpt1 3824 . . . . . . 7 x(x AB)
92, 8nfcxfr 2157 . . . . . 6 x𝐹
10 nfcv 2160 . . . . . 6 xy
117, 9, 10nfbr 3782 . . . . 5 x 𝐷𝐹y
12 nfv 1402 . . . . 5 x y𝐶
1311, 12nfim 1446 . . . 4 x(𝐷𝐹yy𝐶)
14 breq1 3741 . . . . 5 (x = 𝐷 → (x𝐹y𝐷𝐹y))
15 fvmptss2.1 . . . . . 6 (x = 𝐷B = 𝐶)
1615sseq2d 2950 . . . . 5 (x = 𝐷 → (yBy𝐶))
1714, 16imbi12d 223 . . . 4 (x = 𝐷 → ((x𝐹yyB) ↔ (𝐷𝐹yy𝐶)))
18 df-br 3739 . . . . 5 (x𝐹y ↔ ⟨x, y 𝐹)
19 opabid 3968 . . . . . . 7 (⟨x, y {⟨x, y⟩ ∣ (x A y = B)} ↔ (x A y = B))
20 eqimss 2974 . . . . . . . 8 (y = ByB)
2120adantl 262 . . . . . . 7 ((x A y = B) → yB)
2219, 21sylbi 114 . . . . . 6 (⟨x, y {⟨x, y⟩ ∣ (x A y = B)} → yB)
23 df-mpt 3794 . . . . . . 7 (x AB) = {⟨x, y⟩ ∣ (x A y = B)}
242, 23eqtri 2042 . . . . . 6 𝐹 = {⟨x, y⟩ ∣ (x A y = B)}
2522, 24eleq2s 2114 . . . . 5 (⟨x, y 𝐹yB)
2618, 25sylbi 114 . . . 4 (x𝐹yyB)
277, 13, 17, 26vtoclgf 2589 . . 3 (𝐷 V → (𝐷𝐹yy𝐶))
286, 27mpcom 32 . 2 (𝐷𝐹yy𝐶)
291, 28mpg 1320 1 (𝐹𝐷) ⊆ 𝐶
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  Vcvv 2535   ⊆ wss 2894  ⟨cop 3353   class class class wbr 3738  {copab 3791   ↦ cmpt 3792  Rel wrel 4277  Fun wfun 4823  ‘cfv 4829 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-iota 4794  df-fun 4831  df-fv 4837 This theorem is referenced by:  mptfvex  5181
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