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Theorem fvmptss2 5279
 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 = (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
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvss 5220 . 2 (∀𝑦(𝐷𝐹𝑦𝑦𝐶) → (𝐹𝐷) ⊆ 𝐶)
2 fvmptss2.2 . . . . . 6 𝐹 = (𝑥𝐴𝐵)
32funmpt2 4969 . . . . 5 Fun 𝐹
4 funrel 4949 . . . . 5 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 7 . . . 4 Rel 𝐹
65brrelexi 4410 . . 3 (𝐷𝐹𝑦𝐷 ∈ V)
7 nfcv 2220 . . . 4 𝑥𝐷
8 nfmpt1 3879 . . . . . . 7 𝑥(𝑥𝐴𝐵)
92, 8nfcxfr 2217 . . . . . 6 𝑥𝐹
10 nfcv 2220 . . . . . 6 𝑥𝑦
117, 9, 10nfbr 3837 . . . . 5 𝑥 𝐷𝐹𝑦
12 nfv 1462 . . . . 5 𝑥 𝑦𝐶
1311, 12nfim 1505 . . . 4 𝑥(𝐷𝐹𝑦𝑦𝐶)
14 breq1 3796 . . . . 5 (𝑥 = 𝐷 → (𝑥𝐹𝑦𝐷𝐹𝑦))
15 fvmptss2.1 . . . . . 6 (𝑥 = 𝐷𝐵 = 𝐶)
1615sseq2d 3028 . . . . 5 (𝑥 = 𝐷 → (𝑦𝐵𝑦𝐶))
1714, 16imbi12d 232 . . . 4 (𝑥 = 𝐷 → ((𝑥𝐹𝑦𝑦𝐵) ↔ (𝐷𝐹𝑦𝑦𝐶)))
18 df-br 3794 . . . . 5 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
19 opabid 4020 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ (𝑥𝐴𝑦 = 𝐵))
20 eqimss 3052 . . . . . . . 8 (𝑦 = 𝐵𝑦𝐵)
2120adantl 271 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → 𝑦𝐵)
2219, 21sylbi 119 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} → 𝑦𝐵)
23 df-mpt 3849 . . . . . . 7 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
242, 23eqtri 2102 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2522, 24eleq2s 2174 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵)
2618, 25sylbi 119 . . . 4 (𝑥𝐹𝑦𝑦𝐵)
277, 13, 17, 26vtoclgf 2658 . . 3 (𝐷 ∈ V → (𝐷𝐹𝑦𝑦𝐶))
286, 27mpcom 36 . 2 (𝐷𝐹𝑦𝑦𝐶)
291, 28mpg 1381 1 (𝐹𝐷) ⊆ 𝐶
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434  Vcvv 2602   ⊆ wss 2974  ⟨cop 3409   class class class wbr 3793  {copab 3846   ↦ cmpt 3847  Rel wrel 4376  Fun wfun 4926  ‘cfv 4932 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-iota 4897  df-fun 4934  df-fv 4940 This theorem is referenced by:  mptfvex  5288
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