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Theorem mptfvex 5506
 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 3006 . . 3 (𝑦 = 𝐶𝑦 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
2 fvmpt2.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
3 nfcv 2281 . . . . 5 𝑦𝐵
4 nfcsb1v 3035 . . . . 5 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3012 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 4023 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
72, 6eqtri 2160 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
81, 7fvmptss2 5496 . 2 (𝐹𝐶) ⊆ 𝐶 / 𝑥𝐵
9 elex 2697 . . . . . 6 (𝐵𝑉𝐵 ∈ V)
109alimi 1431 . . . . 5 (∀𝑥 𝐵𝑉 → ∀𝑥 𝐵 ∈ V)
113nfel1 2292 . . . . . 6 𝑦 𝐵 ∈ V
124nfel1 2292 . . . . . 6 𝑥𝑦 / 𝑥𝐵 ∈ V
135eleq1d 2208 . . . . . 6 (𝑥 = 𝑦 → (𝐵 ∈ V ↔ 𝑦 / 𝑥𝐵 ∈ V))
1411, 12, 13cbval 1727 . . . . 5 (∀𝑥 𝐵 ∈ V ↔ ∀𝑦𝑦 / 𝑥𝐵 ∈ V)
1510, 14sylib 121 . . . 4 (∀𝑥 𝐵𝑉 → ∀𝑦𝑦 / 𝑥𝐵 ∈ V)
161eleq1d 2208 . . . . 5 (𝑦 = 𝐶 → (𝑦 / 𝑥𝐵 ∈ V ↔ 𝐶 / 𝑥𝐵 ∈ V))
1716spcgv 2773 . . . 4 (𝐶𝑊 → (∀𝑦𝑦 / 𝑥𝐵 ∈ V → 𝐶 / 𝑥𝐵 ∈ V))
1815, 17syl5 32 . . 3 (𝐶𝑊 → (∀𝑥 𝐵𝑉𝐶 / 𝑥𝐵 ∈ V))
1918impcom 124 . 2 ((∀𝑥 𝐵𝑉𝐶𝑊) → 𝐶 / 𝑥𝐵 ∈ V)
20 ssexg 4067 . 2 (((𝐹𝐶) ⊆ 𝐶 / 𝑥𝐵𝐶 / 𝑥𝐵 ∈ V) → (𝐹𝐶) ∈ V)
218, 19, 20sylancr 410 1 ((∀𝑥 𝐵𝑉𝐶𝑊) → (𝐹𝐶) ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1329   = wceq 1331   ∈ wcel 1480  Vcvv 2686  ⦋csb 3003   ⊆ wss 3071   ↦ cmpt 3989  ‘cfv 5123 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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-iota 5088  df-fun 5125  df-fv 5131 This theorem is referenced by:  mpofvex  6101  xpcomco  6720
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