Theorem lmfss 21898

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 21897	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ∈ (𝑋 ↑pm ℂ))
2		toponmax 21528	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
3		cnex 10612	. . . . 5 ⊢ ℂ ∈ V
4		elpmg 8416	. . . . 5 ⊢ ((𝑋 ∈ 𝐽 ∧ ℂ ∈ V) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))))
5	2, 3, 4	sylancl 588	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))))
6	5	adantr 483	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → (𝐹 ∈ (𝑋 ↑pm ℂ) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋))))
7	1, 6	mpbid 234	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → (Fun 𝐹 ∧ 𝐹 ⊆ (ℂ × 𝑋)))
8	7	simprd 498	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝑋))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  Vcvv 3495   ⊆ wss 3936   class class class wbr 5059   × cxp 5548  Fun wfun 6344  ‘cfv 6350  (class class class)co 7150   ↑_pm cpm 8401  ℂcc 10529  TopOnctopon 21512  ⇝_𝑡clm 21828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-pm 8403  df-top 21496  df-topon 21513  df-lm 21831

This theorem is referenced by:  lmss  21900