Theorem lmfss 15328

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(⇝𝑡‘𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋))

Proof of Theorem lmfss

Step	Hyp	Ref	Expression
1		lmfpm 15327	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ∈ (𝑋 ↑pm ℂ))
2		toponmax 15109	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
3		cnex 8297	. . . . 5 ⊢ ℂ ∈ V
4		elpmg 6932	. . . . 5 ⊢ ((𝑋 ∈ 𝐽 ∧ ℂ ∈ V) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))))
5	2, 3, 4	sylancl 417	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))))
6	5	adantr 276	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))))
7	1, 6	mpbid 147	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋)))
8	7	simprd 114	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220   class class class wbr 4128   × cxp 4770  Fun wfun 5369  ‘cfv 5375  (class class class)co 6079   ↑_pm cpm 6917  ℂcc 8171  TopOnctopon 15094  ⇝_𝑡clm 15271

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-pm 6919  df-top 15082  df-topon 15095  df-lm 15274

This theorem is referenced by:  lmss  15330