Theorem cnpf2 21852

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 2821	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2		eqid 2821	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnpf 21849	. . 3 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:∪ 𝐽⟶∪ 𝐾)
4		toponuni 21516	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
5	4	feq2d 6494	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶𝑌))
6		toponuni 21516	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾)
7	6	feq3d 6495	. . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → (𝐹:∪ 𝐽⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾))
8	5, 7	sylan9bb 512	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾))
9	3, 8	syl5ibr 248	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶𝑌))
10	9	3impia 1113	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2110  ∪ cuni 4831  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  TopOnctopon 21512   CnP ccnp 21827

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 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

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-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402  df-top 21496  df-topon 21513  df-cnp 21830

This theorem is referenced by:  iscnp4  21865  1stccnp  22064  txcnp  22222  ptcnplem  22223  ptcnp  22224  cnpflf2  22602  cnpflf  22603  flfcnp  22606  flfcnp2  22609  cnpfcf  22643  ghmcnp  22717  metcnpi3  23150  limcvallem  24463  cnplimc  24479  limccnp  24483  limccnp2  24484  ftc1lem3  24629