Theorem txomap 33780

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 786	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝜑)
2		simpllr 775	. . . . . . 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 583	. . . . . 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 768	. . . . . . 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 583	. . . . . 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 5484	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
10	8, 9	fnmpoi 8111	. . . . . . . 8 ⊢ 𝐻 Fn (𝑋 × 𝑌)
11		txomap.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
12	11	ad6antr 735	. . . . . . . . . 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 22954	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
14	12, 2, 13	syl2anc 583	. . . . . . . . 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 735	. . . . . . . . . 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 22954	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦 ∈ 𝐾) → 𝑦 ⊆ 𝑌)
18	16, 5, 17	syl2anc 583	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑦 ⊆ 𝑌)
19		xpss12 5715	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
20	14, 18, 19	syl2anc 583	. . . . . . . 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 770	. . . . . . . 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 7270	. . . . . . . 8 ⊢ ((𝐻 Fn (𝑋 × 𝑌) ∧ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑥 × 𝑦)) → (𝐻‘𝑧) ∈ (𝐻 “ (𝑥 × 𝑦)))
23	10, 20, 21, 22	mp3an2i 1466	. . . . . . 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 783	. . . . . . 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 6747	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑍 → 𝐹 Fn 𝑋)
27	1, 25, 26	3syl 18	. . . . . . . 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 6747	. . . . . . . . 9 ⊢ (𝐺:𝑌⟶𝑇 → 𝐺 Fn 𝑌)
30	1, 28, 29	3syl 18	. . . . . . . 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 8176	. . . . . . 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 2858	. . . . . 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 6132	. . . . . . . 8 ⊢ ((𝑥 × 𝑦) ⊆ 𝐴 → (𝐻 “ (𝑥 × 𝑦)) ⊆ (𝐻 “ 𝐴))
34	33	ad2antll 728	. . . . . . 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 4048	. . . . . 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 5714	. . . . . . . . 9 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑎 × 𝑏) = ((𝐹 “ 𝑥) × 𝑏))
37	36	eleq2d 2830	. . . . . . . 8 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑐 ∈ (𝑎 × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏)))
38	36	sseq1d 4040	. . . . . . . 8 ⊢ (𝑎 = (𝐹 “ 𝑥) → ((𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴) ↔ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴)))
39	37, 38	anbi12d 631	. . . . . . 7 ⊢ (𝑎 = (𝐹 “ 𝑥) → ((𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)) ↔ (𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ∧ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴))))
40		xpeq2 5721	. . . . . . . . 9 ⊢ (𝑏 = (𝐺 “ 𝑦) → ((𝐹 “ 𝑥) × 𝑏) = ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)))
41	40	eleq2d 2830	. . . . . . . 8 ⊢ (𝑏 = (𝐺 “ 𝑦) → (𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦))))
42	40	sseq1d 4040	. . . . . . . 8 ⊢ (𝑏 = (𝐺 “ 𝑦) → (((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴) ↔ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ⊆ (𝐻 “ 𝐴)))
43	41, 42	anbi12d 631	. . . . . . 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 3648	. . . . . 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 1374	. . . . 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 23597	. . . . . . . . 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 583	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)))
49	46, 48	mpbid 232	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
50	49	r19.21bi 3257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
51	50	ad4ant13 750	. . . . 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 7574	. . . . . 6 ⊢ Fun 𝐻
54		fvelima 6987	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
55	53, 54	mpan 689	. . . . 5 ⊢ (𝑐 ∈ (𝐻 “ 𝐴) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
56	55	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
57	52, 56	r19.29a 3168	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
58	57	ralrimiva 3152	. 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 23597	. . 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 583	. 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 257	1 ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ⟨cop 4654   × cxp 5698   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  TopOnctopon 22937   ×_t ctx 23589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-topgen 17503  df-topon 22938  df-tx 23591

This theorem is referenced by:  qtophaus  33782