Theorem iscn 21039

Description: The predicate "𝐹 is a continuous function from topology 𝐽 to topology 𝐾." Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
iscn	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))

Distinct variable groups:   𝑦,𝐽   𝑦,𝐾   𝑦,𝑋   𝑦,𝐹   𝑦,𝑌

Proof of Theorem iscn

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnfval 21037	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
2	1	eleq2d 2687	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽}))
3		cnveq 5296	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
4	3	imaeq1d 5465	. . . . . 6 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑦) = (◡𝐹 “ 𝑦))
5	4	eleq1d 2686	. . . . 5 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑦) ∈ 𝐽 ↔ (◡𝐹 “ 𝑦) ∈ 𝐽))
6	5	ralbidv 2986	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))
7	6	elrab 3363	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌 ↑𝑚 𝑋) ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))
8		toponmax 20730	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
9		toponmax 20730	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
10		elmapg 7870	. . . . 5 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑋 ∈ 𝐽) → (𝐹 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝐹:𝑋⟶𝑌))
11	8, 9, 10	syl2anr 495	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝐹:𝑋⟶𝑌))
12	11	anbi1d 741	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹 ∈ (𝑌 ↑𝑚 𝑋) ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
13	7, 12	syl5bb 272	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
14	2, 13	bitrd 268	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  ^◡ccnv 5113   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031

This theorem is referenced by:  iscn2  21042  cnf2  21053  tgcn  21056  ssidcn  21059  iscncl  21073  cnntr  21079  cnss1  21080  cnss2  21081  cncnp  21084  cnrest  21089  cnrest2  21090  cndis  21095  cnindis  21096  kgencn  21359  kgencn3  21361  tx1cn  21412  tx2cn  21413  txdis1cn  21438  qtopid  21508  qtopcn  21517  qtopf1  21619  qustgplem  21924  ucncn  22089  cvmlift2lem9a  31285  rfcnpre1  39178  0cnf  40090