Theorem txomap 33833

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 787	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝜑)
2		simpllr 776	. . . . . . 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 584	. . . . . 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 769	. . . . . . 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 584	. . . . . 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 5469	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
10	8, 9	fnmpoi 8095	. . . . . . . 8 ⊢ 𝐻 Fn (𝑋 × 𝑌)
11		txomap.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
12	11	ad6antr 736	. . . . . . . . . 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 22933	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
14	12, 2, 13	syl2anc 584	. . . . . . . . 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 736	. . . . . . . . . 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 22933	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦 ∈ 𝐾) → 𝑦 ⊆ 𝑌)
18	16, 5, 17	syl2anc 584	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑦 ⊆ 𝑌)
19		xpss12 5700	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
20	14, 18, 19	syl2anc 584	. . . . . . . 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 771	. . . . . . . 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 7253	. . . . . . . 8 ⊢ ((𝐻 Fn (𝑋 × 𝑌) ∧ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑥 × 𝑦)) → (𝐻‘𝑧) ∈ (𝐻 “ (𝑥 × 𝑦)))
23	10, 20, 21, 22	mp3an2i 1468	. . . . . . 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 784	. . . . . . 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 6736	. . . . . . . . 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 6736	. . . . . . . . 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 8160	. . . . . . 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 2855	. . . . . 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 6120	. . . . . . . 8 ⊢ ((𝑥 × 𝑦) ⊆ 𝐴 → (𝐻 “ (𝑥 × 𝑦)) ⊆ (𝐻 “ 𝐴))
34	33	ad2antll 729	. . . . . . 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 4019	. . . . . 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 5699	. . . . . . . . 9 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑎 × 𝑏) = ((𝐹 “ 𝑥) × 𝑏))
37	36	eleq2d 2827	. . . . . . . 8 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑐 ∈ (𝑎 × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏)))
38	36	sseq1d 4015	. . . . . . . 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 5706	. . . . . . . . 9 ⊢ (𝑏 = (𝐺 “ 𝑦) → ((𝐹 “ 𝑥) × 𝑏) = ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)))
41	40	eleq2d 2827	. . . . . . . 8 ⊢ (𝑏 = (𝐺 “ 𝑦) → (𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦))))
42	40	sseq1d 4015	. . . . . . . 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 1376	. . . . 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 23576	. . . . . . . . 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 584	. . . . . . . 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 3251	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
51	50	ad4ant13 751	. . . . 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 3216	. . . 4 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
53	8	mpofun 7557	. . . . . 6 ⊢ Fun 𝐻
54		fvelima 6974	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
55	53, 54	mpan 690	. . . . 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 3162	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
58	57	ralrimiva 3146	. 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 23576	. . 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 584	. 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 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ⟨cop 4632   × cxp 5683   “ cima 5688  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  TopOnctopon 22916   ×_t ctx 23568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-topgen 17488  df-topon 22917  df-tx 23570

This theorem is referenced by:  qtophaus  33835