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Theorem lmfss 20849
Description: Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition). (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
Assertion
Ref Expression
lmfss ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋))

Proof of Theorem lmfss
StepHypRef Expression
1 lmfpm 20848 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ∈ (𝑋pm ℂ))
2 toponmax 20482 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
3 cnex 9870 . . . . 5 ℂ ∈ V
4 elpmg 7733 . . . . 5 ((𝑋𝐽 ∧ ℂ ∈ V) → (𝐹 ∈ (𝑋pm ℂ) ↔ (Fun 𝐹𝐹 ⊆ (ℂ × 𝑋))))
52, 3, 4sylancl 692 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝐹 ∈ (𝑋pm ℂ) ↔ (Fun 𝐹𝐹 ⊆ (ℂ × 𝑋))))
65adantr 479 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → (𝐹 ∈ (𝑋pm ℂ) ↔ (Fun 𝐹𝐹 ⊆ (ℂ × 𝑋))))
71, 6mpbid 220 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → (Fun 𝐹𝐹 ⊆ (ℂ × 𝑋)))
87simprd 477 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  Vcvv 3169  wss 3536   class class class wbr 4574   × cxp 5023  Fun wfun 5781  cfv 5787  (class class class)co 6524  pm cpm 7719  cc 9787  TopOnctopon 20457  𝑡clm 20779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-pm 7721  df-top 20460  df-topon 20462  df-lm 20782
This theorem is referenced by:  lmss  20851
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