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Theorem mptfvex 5336
Description: Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
Hypothesis
Ref Expression
fvmpt2.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
mptfvex ((∀𝑥 𝐵𝑉𝐶𝑊) → (𝐹𝐶) ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem mptfvex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2924 . . 3 (𝑦 = 𝐶𝑦 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
2 fvmpt2.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
3 nfcv 2225 . . . . 5 𝑦𝐵
4 nfcsb1v 2951 . . . . 5 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 2929 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 3901 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
72, 6eqtri 2105 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
81, 7fvmptss2 5327 . 2 (𝐹𝐶) ⊆ 𝐶 / 𝑥𝐵
9 elex 2623 . . . . . 6 (𝐵𝑉𝐵 ∈ V)
109alimi 1387 . . . . 5 (∀𝑥 𝐵𝑉 → ∀𝑥 𝐵 ∈ V)
113nfel1 2235 . . . . . 6 𝑦 𝐵 ∈ V
124nfel1 2235 . . . . . 6 𝑥𝑦 / 𝑥𝐵 ∈ V
135eleq1d 2153 . . . . . 6 (𝑥 = 𝑦 → (𝐵 ∈ V ↔ 𝑦 / 𝑥𝐵 ∈ V))
1411, 12, 13cbval 1681 . . . . 5 (∀𝑥 𝐵 ∈ V ↔ ∀𝑦𝑦 / 𝑥𝐵 ∈ V)
1510, 14sylib 120 . . . 4 (∀𝑥 𝐵𝑉 → ∀𝑦𝑦 / 𝑥𝐵 ∈ V)
161eleq1d 2153 . . . . 5 (𝑦 = 𝐶 → (𝑦 / 𝑥𝐵 ∈ V ↔ 𝐶 / 𝑥𝐵 ∈ V))
1716spcgv 2698 . . . 4 (𝐶𝑊 → (∀𝑦𝑦 / 𝑥𝐵 ∈ V → 𝐶 / 𝑥𝐵 ∈ V))
1815, 17syl5 32 . . 3 (𝐶𝑊 → (∀𝑥 𝐵𝑉𝐶 / 𝑥𝐵 ∈ V))
1918impcom 123 . 2 ((∀𝑥 𝐵𝑉𝐶𝑊) → 𝐶 / 𝑥𝐵 ∈ V)
20 ssexg 3946 . 2 (((𝐹𝐶) ⊆ 𝐶 / 𝑥𝐵𝐶 / 𝑥𝐵 ∈ V) → (𝐹𝐶) ∈ V)
218, 19, 20sylancr 405 1 ((∀𝑥 𝐵𝑉𝐶𝑊) → (𝐹𝐶) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1285   = wceq 1287  wcel 1436  Vcvv 2614  csb 2921  wss 2986  cmpt 3868  cfv 4972
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-iota 4937  df-fun 4974  df-fv 4980
This theorem is referenced by:  mpt2fvex  5911  xpcomco  6475
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