Theorem iscnp3 23577

Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". (Contributed by NM, 15-May-2007.)

Assertion

Ref	Expression
iscnp3	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))

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

Proof of Theorem iscnp3

Step	Hyp	Ref	Expression
1		iscnp 23570	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
2		ffun 6723	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
3	2	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → Fun 𝐹)
4		toponss 23260	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
5	4	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
6		fdm 6730	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
7	6	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → dom 𝐹 = 𝑋)
8	5, 7	sseqtrrd 3950	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ dom 𝐹)
9		funimass3 7064	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ⊆ dom 𝐹) → ((𝐹 “ 𝑥) ⊆ 𝑦 ↔ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
10	3, 8, 9	syl2anc 596	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝑥) ⊆ 𝑦 ↔ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
11	10	anbi2d 642	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝐽) → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
12	11	rexbidva 3188	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
13	12	imbi2d 343	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → (((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ↔ ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))))
14	13	ralbidv 3189	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ↔ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))))
15	14	pm5.32da 590	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))
16	15	3ad2ant1 1151	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))
17	1, 16	bitrd 282	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ^◡ccnv 5654  dom cdm 5655   “ cima 5658  Fun wfun 6541  ⟶wf 6543  ‘cfv 6547  (class class class)co 7430  TopOnctopon 23243   CnP ccnp 23558

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

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 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8863  df-top 23227  df-topon 23244  df-cnp 23561

This theorem is used by:  cncnpi  23611  cnpdis  23626