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Theorem mptfvex 5199
 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 = (x AB)
Assertion
Ref Expression
mptfvex ((x B 𝑉 𝐶 𝑊) → (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 2849 . . 3 (y = 𝐶y / xB = 𝐶 / xB)
2 fvmpt2.1 . . . 4 𝐹 = (x AB)
3 nfcv 2175 . . . . 5 yB
4 nfcsb1v 2876 . . . . 5 xy / xB
5 csbeq1a 2854 . . . . 5 (x = yB = y / xB)
63, 4, 5cbvmpt 3842 . . . 4 (x AB) = (y Ay / xB)
72, 6eqtri 2057 . . 3 𝐹 = (y Ay / xB)
81, 7fvmptss2 5190 . 2 (𝐹𝐶) ⊆ 𝐶 / xB
9 elex 2560 . . . . . 6 (B 𝑉B V)
109alimi 1341 . . . . 5 (x B 𝑉x B V)
113nfel1 2185 . . . . . 6 y B V
124nfel1 2185 . . . . . 6 xy / xB V
135eleq1d 2103 . . . . . 6 (x = y → (B V ↔ y / xB V))
1411, 12, 13cbval 1634 . . . . 5 (x B V ↔ yy / xB V)
1510, 14sylib 127 . . . 4 (x B 𝑉yy / xB V)
161eleq1d 2103 . . . . 5 (y = 𝐶 → (y / xB V ↔ 𝐶 / xB V))
1716spcgv 2634 . . . 4 (𝐶 𝑊 → (yy / xB V → 𝐶 / xB V))
1815, 17syl5 28 . . 3 (𝐶 𝑊 → (x B 𝑉𝐶 / xB V))
1918impcom 116 . 2 ((x B 𝑉 𝐶 𝑊) → 𝐶 / xB V)
20 ssexg 3887 . 2 (((𝐹𝐶) ⊆ 𝐶 / xB 𝐶 / xB V) → (𝐹𝐶) V)
218, 19, 20sylancr 393 1 ((x B 𝑉 𝐶 𝑊) → (𝐹𝐶) V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ⦋csb 2846   ⊆ wss 2911   ↦ cmpt 3809  ‘cfv 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-iota 4810  df-fun 4847  df-fv 4853 This theorem is referenced by:  mpt2fvex  5771  xpcomco  6236
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