Theorem txomap 30442

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 809	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝜑)
2		simpllr 793	. . . . . . 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 579	. . . . . 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 785	. . . . . . 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 579	. . . . . 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	. . . . . . . . . 10 ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
9		opex 5155	. . . . . . . . . 10 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
10	8, 9	fnmpt2i 7507	. . . . . . . . 9 ⊢ 𝐻 Fn (𝑋 × 𝑌)
11	10	a1i 11	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝐻 Fn (𝑋 × 𝑌))
12		txomap.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
13	1, 12	syl 17	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
14		toponss 21109	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
15	13, 2, 14	syl2anc 579	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑥 ⊆ 𝑋)
16		txomap.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
17	1, 16	syl 17	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝐾 ∈ (TopOn‘𝑌))
18		toponss 21109	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦 ∈ 𝐾) → 𝑦 ⊆ 𝑌)
19	17, 5, 18	syl2anc 579	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑦 ⊆ 𝑌)
20		xpss12 5361	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
21	15, 19, 20	syl2anc 579	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
22		simprl 787	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → 𝑧 ∈ (𝑥 × 𝑦))
23		fnfvima 6757	. . . . . . . 8 ⊢ ((𝐻 Fn (𝑋 × 𝑌) ∧ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑥 × 𝑦)) → (𝐻‘𝑧) ∈ (𝐻 “ (𝑥 × 𝑦)))
24	11, 21, 22, 23	syl3anc 1494	. . . . . . 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)))
25		simp-4r 803	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐾) ∧ (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)) → (𝐻‘𝑧) = 𝑐)
26		txomap.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋⟶𝑍)
27		ffn 6282	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑍 → 𝐹 Fn 𝑋)
28	1, 26, 27	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)
29		txomap.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑌⟶𝑇)
30		ffn 6282	. . . . . . . . 9 ⊢ (𝐺:𝑌⟶𝑇 → 𝐺 Fn 𝑌)
31	1, 29, 30	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)
32	8, 28, 31, 15, 19	fimaproj 30441	. . . . . . 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)))
33	24, 25, 32	3eltr3d 2920	. . . . . 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)))
34		imass2 5746	. . . . . . . 8 ⊢ ((𝑥 × 𝑦) ⊆ 𝐴 → (𝐻 “ (𝑥 × 𝑦)) ⊆ (𝐻 “ 𝐴))
35	34	ad2antll 720	. . . . . . 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))
36	32, 35	eqsstr3d 3865	. . . . . 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))
37		xpeq1 5360	. . . . . . . . 9 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑎 × 𝑏) = ((𝐹 “ 𝑥) × 𝑏))
38	37	eleq2d 2892	. . . . . . . 8 ⊢ (𝑎 = (𝐹 “ 𝑥) → (𝑐 ∈ (𝑎 × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏)))
39	37	sseq1d 3857	. . . . . . . 8 ⊢ (𝑎 = (𝐹 “ 𝑥) → ((𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴) ↔ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴)))
40	38, 39	anbi12d 624	. . . . . . 7 ⊢ (𝑎 = (𝐹 “ 𝑥) → ((𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)) ↔ (𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ∧ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴))))
41		xpeq2 5367	. . . . . . . . 9 ⊢ (𝑏 = (𝐺 “ 𝑦) → ((𝐹 “ 𝑥) × 𝑏) = ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)))
42	41	eleq2d 2892	. . . . . . . 8 ⊢ (𝑏 = (𝐺 “ 𝑦) → (𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ↔ 𝑐 ∈ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦))))
43	41	sseq1d 3857	. . . . . . . 8 ⊢ (𝑏 = (𝐺 “ 𝑦) → (((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴) ↔ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ⊆ (𝐻 “ 𝐴)))
44	42, 43	anbi12d 624	. . . . . . 7 ⊢ (𝑏 = (𝐺 “ 𝑦) → ((𝑐 ∈ ((𝐹 “ 𝑥) × 𝑏) ∧ ((𝐹 “ 𝑥) × 𝑏) ⊆ (𝐻 “ 𝐴)) ↔ (𝑐 ∈ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ∧ ((𝐹 “ 𝑥) × (𝐺 “ 𝑦)) ⊆ (𝐻 “ 𝐴))))
45	40, 44	rspc2ev 3541	. . . . . 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)))
46	4, 7, 33, 36, 45	syl112anc 1497	. . . . 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)))
47		txomap.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ×t 𝐾))
48		eltx 21749	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐴 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)))
49	12, 16, 48	syl2anc 579	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴)))
50	47, 49	mpbid 224	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
51	50	r19.21bi 3141	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
52	51	adantlr 706	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
53	52	adantr 474	. . . . 5 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑧 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝐴))
54	46, 53	r19.29vva 3291	. . . 4 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) ∧ 𝑧 ∈ 𝐴) ∧ (𝐻‘𝑧) = 𝑐) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
55	8	mpt2fun 7027	. . . . . 6 ⊢ Fun 𝐻
56		fvelima 6499	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
57	55, 56	mpan 681	. . . . 5 ⊢ (𝑐 ∈ (𝐻 “ 𝐴) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
58	57	adantl 475	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑧 ∈ 𝐴 (𝐻‘𝑧) = 𝑐)
59	54, 58	r19.29a 3288	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐻 “ 𝐴)) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
60	59	ralrimiva 3175	. 2 ⊢ (𝜑 → ∀𝑐 ∈ (𝐻 “ 𝐴)∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴)))
61		txomap.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
62		txomap.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑇))
63		eltx 21749	. . 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))))
64	61, 62, 63	syl2anc 579	. 2 ⊢ (𝜑 → ((𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀) ↔ ∀𝑐 ∈ (𝐻 “ 𝐴)∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑀 (𝑐 ∈ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐻 “ 𝐴))))
65	60, 64	mpbird 249	1 ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118   ⊆ wss 3798  ⟨cop 4405   × cxp 5344   “ cima 5349  Fun wfun 6121   Fn wfn 6122  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910   ↦ cmpt2 6912  TopOnctopon 21092   ×_t ctx 21741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-topgen 16464  df-topon 21093  df-tx 21743

This theorem is referenced by:  qtophaus  30444