ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mptfvex GIF version

Theorem mptfvex 5614
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 3072 . . 3 (𝑦 = 𝐶𝑦 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
2 fvmpt2.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
3 nfcv 2329 . . . . 5 𝑦𝐵
4 nfcsb1v 3102 . . . . 5 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3078 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 4110 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
72, 6eqtri 2208 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
81, 7fvmptss2 5604 . 2 (𝐹𝐶) ⊆ 𝐶 / 𝑥𝐵
9 elex 2760 . . . . . 6 (𝐵𝑉𝐵 ∈ V)
109alimi 1465 . . . . 5 (∀𝑥 𝐵𝑉 → ∀𝑥 𝐵 ∈ V)
113nfel1 2340 . . . . . 6 𝑦 𝐵 ∈ V
124nfel1 2340 . . . . . 6 𝑥𝑦 / 𝑥𝐵 ∈ V
135eleq1d 2256 . . . . . 6 (𝑥 = 𝑦 → (𝐵 ∈ V ↔ 𝑦 / 𝑥𝐵 ∈ V))
1411, 12, 13cbval 1764 . . . . 5 (∀𝑥 𝐵 ∈ V ↔ ∀𝑦𝑦 / 𝑥𝐵 ∈ V)
1510, 14sylib 122 . . . 4 (∀𝑥 𝐵𝑉 → ∀𝑦𝑦 / 𝑥𝐵 ∈ V)
161eleq1d 2256 . . . . 5 (𝑦 = 𝐶 → (𝑦 / 𝑥𝐵 ∈ V ↔ 𝐶 / 𝑥𝐵 ∈ V))
1716spcgv 2836 . . . 4 (𝐶𝑊 → (∀𝑦𝑦 / 𝑥𝐵 ∈ V → 𝐶 / 𝑥𝐵 ∈ V))
1815, 17syl5 32 . . 3 (𝐶𝑊 → (∀𝑥 𝐵𝑉𝐶 / 𝑥𝐵 ∈ V))
1918impcom 125 . 2 ((∀𝑥 𝐵𝑉𝐶𝑊) → 𝐶 / 𝑥𝐵 ∈ V)
20 ssexg 4154 . 2 (((𝐹𝐶) ⊆ 𝐶 / 𝑥𝐵𝐶 / 𝑥𝐵 ∈ V) → (𝐹𝐶) ∈ V)
218, 19, 20sylancr 414 1 ((∀𝑥 𝐵𝑉𝐶𝑊) → (𝐹𝐶) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1361   = wceq 1363  wcel 2158  Vcvv 2749  csb 3069  wss 3141  cmpt 4076  cfv 5228
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-iota 5190  df-fun 5230  df-fv 5236
This theorem is referenced by:  mpofvex  6218  xpcomco  6840  lssex  13543
  Copyright terms: Public domain W3C validator