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Theorem mptfvex 5603
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 3062 . . 3 (𝑦 = 𝐶𝑦 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
2 fvmpt2.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
3 nfcv 2319 . . . . 5 𝑦𝐵
4 nfcsb1v 3092 . . . . 5 𝑥𝑦 / 𝑥𝐵
5 csbeq1a 3068 . . . . 5 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
63, 4, 5cbvmpt 4100 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
72, 6eqtri 2198 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
81, 7fvmptss2 5593 . 2 (𝐹𝐶) ⊆ 𝐶 / 𝑥𝐵
9 elex 2750 . . . . . 6 (𝐵𝑉𝐵 ∈ V)
109alimi 1455 . . . . 5 (∀𝑥 𝐵𝑉 → ∀𝑥 𝐵 ∈ V)
113nfel1 2330 . . . . . 6 𝑦 𝐵 ∈ V
124nfel1 2330 . . . . . 6 𝑥𝑦 / 𝑥𝐵 ∈ V
135eleq1d 2246 . . . . . 6 (𝑥 = 𝑦 → (𝐵 ∈ V ↔ 𝑦 / 𝑥𝐵 ∈ V))
1411, 12, 13cbval 1754 . . . . 5 (∀𝑥 𝐵 ∈ V ↔ ∀𝑦𝑦 / 𝑥𝐵 ∈ V)
1510, 14sylib 122 . . . 4 (∀𝑥 𝐵𝑉 → ∀𝑦𝑦 / 𝑥𝐵 ∈ V)
161eleq1d 2246 . . . . 5 (𝑦 = 𝐶 → (𝑦 / 𝑥𝐵 ∈ V ↔ 𝐶 / 𝑥𝐵 ∈ V))
1716spcgv 2826 . . . 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 4144 . 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 1351   = wceq 1353  wcel 2148  Vcvv 2739  csb 3059  wss 3131  cmpt 4066  cfv 5218
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-iota 5180  df-fun 5220  df-fv 5226
This theorem is referenced by:  mpofvex  6206  xpcomco  6828
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