Theorem lmfss 23620

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 23619	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ∈ (𝑋 ↑pm ℂ))
2		toponmax 23250	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
3		cnex 11285	. . . . 5 ⊢ ℂ ∈ V
4		elpmg 8867	. . . . 5 ⊢ ((𝑋 ∈ 𝐽 ∧ ℂ ∈ V) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))))
5	2, 3, 4	sylancl 598	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))))
6	5	adantr 486	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))))
7	1, 6	mpbid 235	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋)))
8	7	simprd 501	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102   × cxp 5648  Fun wfun 6533  ‘cfv 6539  (class class class)co 7421   ↑_pm cpm 8852  ℂcc 11202  TopOnctopon 23234  ⇝_𝑡clm 23550

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-pm 8854  df-top 23218  df-topon 23235  df-lm 23553

This theorem is used by:  lmss  23622