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Theorem fvmptss2 5190
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 5132 . 2 (y(𝐷𝐹yy𝐶) → (𝐹𝐷) ⊆ 𝐶)
2 fvmptss2.2 . . . . . 6 𝐹 = (x AB)
32funmpt2 4882 . . . . 5 Fun 𝐹
4 funrel 4862 . . . . 5 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 7 . . . 4 Rel 𝐹
65brrelexi 4327 . . 3 (𝐷𝐹y𝐷 V)
7 nfcv 2175 . . . 4 x𝐷
8 nfmpt1 3841 . . . . . . 7 x(x AB)
92, 8nfcxfr 2172 . . . . . 6 x𝐹
10 nfcv 2175 . . . . . 6 xy
117, 9, 10nfbr 3799 . . . . 5 x 𝐷𝐹y
12 nfv 1418 . . . . 5 x y𝐶
1311, 12nfim 1461 . . . 4 x(𝐷𝐹yy𝐶)
14 breq1 3758 . . . . 5 (x = 𝐷 → (x𝐹y𝐷𝐹y))
15 fvmptss2.1 . . . . . 6 (x = 𝐷B = 𝐶)
1615sseq2d 2967 . . . . 5 (x = 𝐷 → (yBy𝐶))
1714, 16imbi12d 223 . . . 4 (x = 𝐷 → ((x𝐹yyB) ↔ (𝐷𝐹yy𝐶)))
18 df-br 3756 . . . . 5 (x𝐹y ↔ ⟨x, y 𝐹)
19 opabid 3985 . . . . . . 7 (⟨x, y {⟨x, y⟩ ∣ (x A y = B)} ↔ (x A y = B))
20 eqimss 2991 . . . . . . . 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 3811 . . . . . . 7 (x AB) = {⟨x, y⟩ ∣ (x A y = B)}
242, 23eqtri 2057 . . . . . 6 𝐹 = {⟨x, y⟩ ∣ (x A y = B)}
2522, 24eleq2s 2129 . . . . 5 (⟨x, y 𝐹yB)
2618, 25sylbi 114 . . . 4 (x𝐹yyB)
277, 13, 17, 26vtoclgf 2606 . . 3 (𝐷 V → (𝐷𝐹yy𝐶))
286, 27mpcom 32 . 2 (𝐷𝐹yy𝐶)
291, 28mpg 1337 1 (𝐹𝐷) ⊆ 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  wss 2911  cop 3370   class class class wbr 3755  {copab 3808  cmpt 3809  Rel wrel 4293  Fun wfun 4839  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by:  mptfvex  5199
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