Theorem txomap 34466

Description: Given two open maps 𝐹 and 𝐺, 𝐻 mapping pairs of sets, is also an open map for the product topology. (Contributed by Thierry Arnoux, 29-Dec-2019.)

Hypotheses

Ref	Expression
txomap.f	⊢ (𝜑 → 𝐹:𝑋⟶𝑍)
txomap.g	⊢ (𝜑 → 𝐺:𝑌⟶𝑇)
txomap.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
txomap.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
txomap.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
txomap.m	⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑇))
txomap.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐿)
txomap.2	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 “ 𝑦) ∈ 𝑀)
txomap.a	⊢ (𝜑 → 𝐴 ∈ (𝐽 ×t 𝐾))
txomap.h	⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)

Assertion

Ref	Expression
txomap	⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑀,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑇(𝑥, 𝑦)   𝑍(𝑥, 𝑦)

Proof of Theorem txomap

Dummy variables 𝑎 𝑏 𝑐 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp-6l 799	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝜑)
2		simpllr 788	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑥 ∈ 𝐽)
3		txomap.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐿)
4	1, 2, 3	syl2anc 596	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → (𝐹 “ 𝑥) ∈ 𝐿)
5		simplr 781	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑦 ∈ 𝐾)
6		txomap.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 “ 𝑦) ∈ 𝑀)
7	1, 5, 6	syl2anc 596	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → (𝐺 “ 𝑦) ∈ 𝑀)
8		txomap.h	. . . . . . . . 9 ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
9		opex 5431	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
10	8, 9	fnmpoi 8081	. . . . . . . 8 ⊢ 𝐻 Fn (𝑋 × 𝑌)
11		txomap.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
12	11	ad6antr 749	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
13		toponss 23245	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
14	12, 2, 13	syl2anc 596	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑥 ⊆ 𝑋)
15		txomap.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
16	15	ad6antr 749	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝐾 ∈ (TopOn‘𝑌))
17		toponss 23245	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦 ∈ 𝐾) → 𝑦 ⊆ 𝑌)
18	16, 5, 17	syl2anc 596	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑦 ⊆ 𝑌)
19		xpss12 5665	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
20	14, 18, 19	syl2anc 596	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
21		simprl 783	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑧 ∈ (𝑥 × 𝑦))
22		fnfvima 7239	. . . . . . . 8 ⊢ ((𝐻 Fn (𝑋 × 𝑌) ∧ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑥 × 𝑦)) → (𝐻‘𝑧) ∈ (𝐻 “ (𝑥 × 𝑦)))
23	10, 20, 21, 22	mp3an2i 1495	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → (𝐻‘𝑧) ∈ (𝐻 “ (𝑥 × 𝑦)))
24		simp-4r 796	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → (𝐻‘𝑧) = 𝑐)
25		txomap.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋⟶𝑍)
26		ffn 6709	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑍 → 𝐹 Fn 𝑋)
27	1, 25, 26	3syl 19	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝐹 Fn 𝑋)
28		txomap.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑌⟶𝑇)
29		ffn 6709	. . . . . . . . 9 ⊢ (𝐺:𝑌⟶𝑇 → 𝐺 Fn 𝑌)
30	1, 28, 29	3syl 19	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝐺 Fn 𝑌)
31	8, 27, 30, 14, 18	fimaproj 8152	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → (𝐻 “ (𝑥 × 𝑦)) = ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)))
32	23, 24, 31	3eltr3d 2874	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑐 ∈ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)))
33		imass2 6054	. . . . . . . 8 ⊢ ((𝑥 × 𝑦) ⊆ 𝐴 → (𝐻 “ (𝑥 × 𝑦)) ⊆ (𝐻 “ 𝐴))
34	33	ad2antll 742	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → (𝐻 “ (𝑥 × 𝑦)) ⊆ (𝐻 “ 𝐴))
35	31, 34	eqsstrrd 3943	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ⊆ (𝐻 “ 𝐴))
36		xpeq1 5664	. . . . . . . . 9 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑎 × 𝑏) = ((𝐹 “ 𝑥) × 𝑏))
37	36	eleq2d 2846	. . . . . . . 8 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑐 ∈ (𝑎 × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏)))
38	36	sseq1d 3939	. . . . . . . 8 ⊢ (𝑎 = (𝐹 “ 𝑥) → ((𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴) ↔ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴)))
39	37, 38	anbi12d 644	. . . . . . 7 ⊢ (𝑎 = (𝐹 “ 𝑥) → ((𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)) ↔ (𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ∧ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴))))
40		xpeq2 5671	. . . . . . . . 9 ⊢ (𝑏 = (𝐺 “ 𝑦) → ((𝐹 “ 𝑥) × 𝑏) = ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)))
41	40	eleq2d 2846	. . . . . . . 8 ⊢ (𝑏 = (𝐺 “ 𝑦) → (𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦))))
42	40	sseq1d 3939	. . . . . . . 8 ⊢ (𝑏 = (𝐺 “ 𝑦) → (((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴) ↔ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ⊆ (𝐻 “ 𝐴)))
43	41, 42	anbi12d 644	. . . . . . 7 ⊢ (𝑏 = (𝐺 “ 𝑦) → ((𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ∧ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴)) ↔ (𝑐 ∈ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ∧ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ⊆ (𝐻 “ 𝐴))))
44	39, 43	rspc2ev 3588	. . . . . 6 ⊢ (((𝐹 “ 𝑥) ∈ 𝐿 ∧ (𝐺 “ 𝑦) ∈ 𝑀 ∧ (𝑐 ∈ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ∧ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ⊆ (𝐻 “ 𝐴))) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
45	4, 7, 32, 35, 44	syl112anc 1401	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
46		txomap.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ×t 𝐾))
47		eltx 23887	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐴 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)))
48	11, 15, 47	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)))
49	46, 48	mpbid 235	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
50	49	r19.21bi 3254	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
51	50	ad4ant13 764	. . . . 5 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
52	45, 51	r19.29vva 3222	. . . 4 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
53	8	mpofun 7544	. . . . . 6 ⊢ Fun 𝐻
54		fvelima 6950	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
55	53, 54	mpan 703	. . . . 5 ⊢ (𝑐 ∈ (𝐻 “ 𝐴) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
56	55	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
57	52, 56	r19.29a 3170	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
58	57	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑐 ∈ (𝐻 “ 𝐴)∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
59		txomap.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
60		txomap.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑇))
61		eltx 23887	. . 3 ⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘𝑇)) → ((𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀) ↔ ∀𝑐 ∈ (𝐻 “ 𝐴)∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴))))
62	59, 60, 61	syl2anc 596	. 2 ⊢ (𝜑 → ((𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀) ↔ ∀𝑐 ∈ (𝐻 “ 𝐴)∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴))))
63	58, 62	mpbird 260	1 ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ⟨cop 4589   × cxp 5648   “ cima 5653  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  TopOnctopon 23228   ×_t ctx 23879

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

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 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-topgen 17614  df-topon 23229  df-tx 23881

This theorem is used by:  qtophaus  34468