Theorem txomap 31093

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 785	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝜑)
2		simpllr 774	. . . . . . 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 586	. . . . . 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 767	. . . . . . 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 586	. . . . . 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 5349	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
10	8, 9	fnmpoi 7762	. . . . . . . 8 ⊢ 𝐻 Fn (𝑋 × 𝑌)
11		txomap.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
12	11	ad6antr 734	. . . . . . . . . 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 21529	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
14	12, 2, 13	syl2anc 586	. . . . . . . . 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 734	. . . . . . . . . 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 21529	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦 ∈ 𝐾) → 𝑦 ⊆ 𝑌)
18	16, 5, 17	syl2anc 586	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑦 ⊆ 𝑌)
19		xpss12 5565	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
20	14, 18, 19	syl2anc 586	. . . . . . . 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 769	. . . . . . . 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 6989	. . . . . . . 8 ⊢ ((𝐻 Fn (𝑋 × 𝑌) ∧ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑥 × 𝑦)) → (𝐻‘𝑧) ∈ (𝐻 “ (𝑥 × 𝑦)))
23	10, 20, 21, 22	mp3an2i 1462	. . . . . . 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 782	. . . . . . 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 6509	. . . . . . . . 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 6509	. . . . . . . . 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 7823	. . . . . . 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 2927	. . . . . 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 5960	. . . . . . . 8 ⊢ ((𝑥 × 𝑦) ⊆ 𝐴 → (𝐻 “ (𝑥 × 𝑦)) ⊆ (𝐻 “ 𝐴))
34	33	ad2antll 727	. . . . . . 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 4006	. . . . . 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 5564	. . . . . . . . 9 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑎 × 𝑏) = ((𝐹 “ 𝑥) × 𝑏))
37	36	eleq2d 2898	. . . . . . . 8 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑐 ∈ (𝑎 × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏)))
38	36	sseq1d 3998	. . . . . . . 8 ⊢ (𝑎 = (𝐹 “ 𝑥) → ((𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴) ↔ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴)))
39	37, 38	anbi12d 632	. . . . . . 7 ⊢ (𝑎 = (𝐹 “ 𝑥) → ((𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)) ↔ (𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ∧ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴))))
40		xpeq2 5571	. . . . . . . . 9 ⊢ (𝑏 = (𝐺 “ 𝑦) → ((𝐹 “ 𝑥) × 𝑏) = ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)))
41	40	eleq2d 2898	. . . . . . . 8 ⊢ (𝑏 = (𝐺 “ 𝑦) → (𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦))))
42	40	sseq1d 3998	. . . . . . . 8 ⊢ (𝑏 = (𝐺 “ 𝑦) → (((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴) ↔ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ⊆ (𝐻 “ 𝐴)))
43	41, 42	anbi12d 632	. . . . . . 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 3635	. . . . . 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 1370	. . . . 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 22170	. . . . . . . . 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 586	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)))
49	46, 48	mpbid 234	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
50	49	r19.21bi 3208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
51	50	ad4ant13 749	. . . . 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 3336	. . . 4 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
53	8	mpofun 7270	. . . . . 6 ⊢ Fun 𝐻
54		fvelima 6726	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
55	53, 54	mpan 688	. . . . 5 ⊢ (𝑐 ∈ (𝐻 “ 𝐴) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
56	55	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
57	52, 56	r19.29a 3289	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
58	57	ralrimiva 3182	. 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 22170	. . 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 586	. 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 259	1 ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936  ⟨cop 4567   × cxp 5548   “ cima 5553  Fun wfun 6344   Fn wfn 6345  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150   ∈ cmpo 7152  TopOnctopon 21512   ×_t ctx 22162

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 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  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 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-topgen 16711  df-topon 21513  df-tx 22164

This theorem is referenced by:  qtophaus  31095