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Theorem fvmptss2 5489
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 5428 . 2 (∀𝑦(𝐷𝐹𝑦𝑦𝐶) → (𝐹𝐷) ⊆ 𝐶)
2 fvmptss2.2 . . . . . 6 𝐹 = (𝑥𝐴𝐵)
32funmpt2 5157 . . . . 5 Fun 𝐹
4 funrel 5135 . . . . 5 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 5 . . . 4 Rel 𝐹
65brrelex1i 4577 . . 3 (𝐷𝐹𝑦𝐷 ∈ V)
7 nfcv 2279 . . . 4 𝑥𝐷
8 nfmpt1 4016 . . . . . . 7 𝑥(𝑥𝐴𝐵)
92, 8nfcxfr 2276 . . . . . 6 𝑥𝐹
10 nfcv 2279 . . . . . 6 𝑥𝑦
117, 9, 10nfbr 3969 . . . . 5 𝑥 𝐷𝐹𝑦
12 nfv 1508 . . . . 5 𝑥 𝑦𝐶
1311, 12nfim 1551 . . . 4 𝑥(𝐷𝐹𝑦𝑦𝐶)
14 breq1 3927 . . . . 5 (𝑥 = 𝐷 → (𝑥𝐹𝑦𝐷𝐹𝑦))
15 fvmptss2.1 . . . . . 6 (𝑥 = 𝐷𝐵 = 𝐶)
1615sseq2d 3122 . . . . 5 (𝑥 = 𝐷 → (𝑦𝐵𝑦𝐶))
1714, 16imbi12d 233 . . . 4 (𝑥 = 𝐷 → ((𝑥𝐹𝑦𝑦𝐵) ↔ (𝐷𝐹𝑦𝑦𝐶)))
18 df-br 3925 . . . . 5 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
19 opabid 4174 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ (𝑥𝐴𝑦 = 𝐵))
20 eqimss 3146 . . . . . . . 8 (𝑦 = 𝐵𝑦𝐵)
2120adantl 275 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → 𝑦𝐵)
2219, 21sylbi 120 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} → 𝑦𝐵)
23 df-mpt 3986 . . . . . . 7 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
242, 23eqtri 2158 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2522, 24eleq2s 2232 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵)
2618, 25sylbi 120 . . . 4 (𝑥𝐹𝑦𝑦𝐵)
277, 13, 17, 26vtoclgf 2739 . . 3 (𝐷 ∈ V → (𝐷𝐹𝑦𝑦𝐶))
286, 27mpcom 36 . 2 (𝐷𝐹𝑦𝑦𝐶)
291, 28mpg 1427 1 (𝐹𝐷) ⊆ 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  Vcvv 2681  wss 3066  cop 3525   class class class wbr 3924  {copab 3983  cmpt 3984  Rel wrel 4539  Fun wfun 5112  cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-iota 5083  df-fun 5120  df-fv 5126
This theorem is referenced by:  mptfvex  5499
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