Theorem cnpf2 23193

Description: A continuous function at point 𝑃 is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cnpf2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)

Proof of Theorem cnpf2

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2736	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnpf 23190	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:∪ 𝐽⟶∪ 𝐾)
4		toponuni 22857	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
5	4	feq2d 6697	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶𝑌))
6		toponuni 22857	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
7	6	feq3d 6698	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → (𝐹:∪ 𝐽⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾))
8	5, 7	sylan9bb 509	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾))
9	3, 8	imbitrrid 246	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶𝑌))
10	9	3impia 1117	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  ∪ cuni 4888  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  TopOnctopon 22853   CnP ccnp 23168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-map 8847  df-top 22837  df-topon 22854  df-cnp 23171

This theorem is referenced by:  iscnp4  23206  1stccnp  23405  txcnp  23563  ptcnplem  23564  ptcnp  23565  cnpflf2  23943  cnpflf  23944  flfcnp  23947  flfcnp2  23950  cnpfcf  23984  ghmcnp  24058  metcnpi3  24490  limcvallem  25829  cnplimc  25845  limccnp  25849  limccnp2  25850  ftc1lem3  26002