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Theorem fvmptss2 5571
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 5510 . 2 (∀𝑦(𝐷𝐹𝑦𝑦𝐶) → (𝐹𝐷) ⊆ 𝐶)
2 fvmptss2.2 . . . . . 6 𝐹 = (𝑥𝐴𝐵)
32funmpt2 5237 . . . . 5 Fun 𝐹
4 funrel 5215 . . . . 5 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 5 . . . 4 Rel 𝐹
65brrelex1i 4654 . . 3 (𝐷𝐹𝑦𝐷 ∈ V)
7 nfcv 2312 . . . 4 𝑥𝐷
8 nfmpt1 4082 . . . . . . 7 𝑥(𝑥𝐴𝐵)
92, 8nfcxfr 2309 . . . . . 6 𝑥𝐹
10 nfcv 2312 . . . . . 6 𝑥𝑦
117, 9, 10nfbr 4035 . . . . 5 𝑥 𝐷𝐹𝑦
12 nfv 1521 . . . . 5 𝑥 𝑦𝐶
1311, 12nfim 1565 . . . 4 𝑥(𝐷𝐹𝑦𝑦𝐶)
14 breq1 3992 . . . . 5 (𝑥 = 𝐷 → (𝑥𝐹𝑦𝐷𝐹𝑦))
15 fvmptss2.1 . . . . . 6 (𝑥 = 𝐷𝐵 = 𝐶)
1615sseq2d 3177 . . . . 5 (𝑥 = 𝐷 → (𝑦𝐵𝑦𝐶))
1714, 16imbi12d 233 . . . 4 (𝑥 = 𝐷 → ((𝑥𝐹𝑦𝑦𝐵) ↔ (𝐷𝐹𝑦𝑦𝐶)))
18 df-br 3990 . . . . 5 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
19 opabid 4242 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ (𝑥𝐴𝑦 = 𝐵))
20 eqimss 3201 . . . . . . . 8 (𝑦 = 𝐵𝑦𝐵)
2120adantl 275 . . . . . . 7 ((𝑥𝐴𝑦 = 𝐵) → 𝑦𝐵)
2219, 21sylbi 120 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} → 𝑦𝐵)
23 df-mpt 4052 . . . . . . 7 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
242, 23eqtri 2191 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2522, 24eleq2s 2265 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵)
2618, 25sylbi 120 . . . 4 (𝑥𝐹𝑦𝑦𝐵)
277, 13, 17, 26vtoclgf 2788 . . 3 (𝐷 ∈ V → (𝐷𝐹𝑦𝑦𝐶))
286, 27mpcom 36 . 2 (𝐷𝐹𝑦𝑦𝐶)
291, 28mpg 1444 1 (𝐹𝐷) ⊆ 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  Vcvv 2730  wss 3121  cop 3586   class class class wbr 3989  {copab 4049  cmpt 4050  Rel wrel 4616  Fun wfun 5192  cfv 5198
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-iota 5160  df-fun 5200  df-fv 5206
This theorem is referenced by:  mptfvex  5581
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