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Theorem fvmptss2 5591
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 5529 . 2 (∀𝑦(𝐷𝐹𝑦𝑦𝐶) → (𝐹𝐷) ⊆ 𝐶)
2 fvmptss2.2 . . . . . 6 𝐹 = (𝑥𝐴𝐵)
32funmpt2 5255 . . . . 5 Fun 𝐹
4 funrel 5233 . . . . 5 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 5 . . . 4 Rel 𝐹
65brrelex1i 4669 . . 3 (𝐷𝐹𝑦𝐷 ∈ V)
7 nfcv 2319 . . . 4 𝑥𝐷
8 nfmpt1 4096 . . . . . . 7 𝑥(𝑥𝐴𝐵)
92, 8nfcxfr 2316 . . . . . 6 𝑥𝐹
10 nfcv 2319 . . . . . 6 𝑥𝑦
117, 9, 10nfbr 4049 . . . . 5 𝑥 𝐷𝐹𝑦
12 nfv 1528 . . . . 5 𝑥 𝑦𝐶
1311, 12nfim 1572 . . . 4 𝑥(𝐷𝐹𝑦𝑦𝐶)
14 breq1 4006 . . . . 5 (𝑥 = 𝐷 → (𝑥𝐹𝑦𝐷𝐹𝑦))
15 fvmptss2.1 . . . . . 6 (𝑥 = 𝐷𝐵 = 𝐶)
1615sseq2d 3185 . . . . 5 (𝑥 = 𝐷 → (𝑦𝐵𝑦𝐶))
1714, 16imbi12d 234 . . . 4 (𝑥 = 𝐷 → ((𝑥𝐹𝑦𝑦𝐵) ↔ (𝐷𝐹𝑦𝑦𝐶)))
18 df-br 4004 . . . . 5 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
19 opabid 4257 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ (𝑥𝐴𝑦 = 𝐵))
20 eqimss 3209 . . . . . . . 8 (𝑦 = 𝐵𝑦𝐵)
2120adantl 277 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → 𝑦𝐵)
2219, 21sylbi 121 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} → 𝑦𝐵)
23 df-mpt 4066 . . . . . . 7 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
242, 23eqtri 2198 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2522, 24eleq2s 2272 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵)
2618, 25sylbi 121 . . . 4 (𝑥𝐹𝑦𝑦𝐵)
277, 13, 17, 26vtoclgf 2795 . . 3 (𝐷 ∈ V → (𝐷𝐹𝑦𝑦𝐶))
286, 27mpcom 36 . 2 (𝐷𝐹𝑦𝑦𝐶)
291, 28mpg 1451 1 (𝐹𝐷) ⊆ 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  wss 3129  cop 3595   class class class wbr 4003  {copab 4063  cmpt 4064  Rel wrel 4631  Fun wfun 5210  cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-iota 5178  df-fun 5218  df-fv 5224
This theorem is referenced by:  mptfvex  5601
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