| Step | Hyp | Ref
| Expression |
| 1 | | txcnmpt.1 |
. . . . . . 7
⊢ 𝑊 = ∪
𝑈 |
| 2 | | eqid 2737 |
. . . . . . 7
⊢ ∪ 𝑅 =
∪ 𝑅 |
| 3 | 1, 2 | cnf 23254 |
. . . . . 6
⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝐹:𝑊⟶∪ 𝑅) |
| 4 | 3 | adantr 480 |
. . . . 5
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐹:𝑊⟶∪ 𝑅) |
| 5 | 4 | ffvelcdmda 7104 |
. . . 4
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐹‘𝑥) ∈ ∪ 𝑅) |
| 6 | | eqid 2737 |
. . . . . . 7
⊢ ∪ 𝑆 =
∪ 𝑆 |
| 7 | 1, 6 | cnf 23254 |
. . . . . 6
⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝐺:𝑊⟶∪ 𝑆) |
| 8 | 7 | adantl 481 |
. . . . 5
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐺:𝑊⟶∪ 𝑆) |
| 9 | 8 | ffvelcdmda 7104 |
. . . 4
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝐺‘𝑥) ∈ ∪ 𝑆) |
| 10 | 5, 9 | opelxpd 5724 |
. . 3
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ 𝑥 ∈ 𝑊) → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (∪
𝑅 × ∪ 𝑆)) |
| 11 | | txcnmpt.2 |
. . 3
⊢ 𝐻 = (𝑥 ∈ 𝑊 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉) |
| 12 | 10, 11 | fmptd 7134 |
. 2
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻:𝑊⟶(∪ 𝑅 × ∪ 𝑆)) |
| 13 | 11 | mptpreima 6258 |
. . . . . 6
⊢ (◡𝐻 “ (𝑟 × 𝑠)) = {𝑥 ∈ 𝑊 ∣ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝑟 × 𝑠)} |
| 14 | 4 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐹:𝑊⟶∪ 𝑅) |
| 15 | 14 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐹:𝑊⟶∪ 𝑅) |
| 16 | | ffn 6736 |
. . . . . . . . . . . 12
⊢ (𝐹:𝑊⟶∪ 𝑅 → 𝐹 Fn 𝑊) |
| 17 | | elpreima 7078 |
. . . . . . . . . . . 12
⊢ (𝐹 Fn 𝑊 → (𝑥 ∈ (◡𝐹 “ 𝑟) ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟))) |
| 18 | 15, 16, 17 | 3syl 18 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (◡𝐹 “ 𝑟) ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟))) |
| 19 | | ibar 528 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝑊 → ((𝐹‘𝑥) ∈ 𝑟 ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟))) |
| 20 | 19 | adantl 481 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → ((𝐹‘𝑥) ∈ 𝑟 ↔ (𝑥 ∈ 𝑊 ∧ (𝐹‘𝑥) ∈ 𝑟))) |
| 21 | 18, 20 | bitr4d 282 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (◡𝐹 “ 𝑟) ↔ (𝐹‘𝑥) ∈ 𝑟)) |
| 22 | 8 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → 𝐺:𝑊⟶∪ 𝑆) |
| 23 | | ffn 6736 |
. . . . . . . . . . . 12
⊢ (𝐺:𝑊⟶∪ 𝑆 → 𝐺 Fn 𝑊) |
| 24 | | elpreima 7078 |
. . . . . . . . . . . 12
⊢ (𝐺 Fn 𝑊 → (𝑥 ∈ (◡𝐺 “ 𝑠) ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠))) |
| 25 | 22, 23, 24 | 3syl 18 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (◡𝐺 “ 𝑠) ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠))) |
| 26 | | ibar 528 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝑊 → ((𝐺‘𝑥) ∈ 𝑠 ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠))) |
| 27 | 26 | adantl 481 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → ((𝐺‘𝑥) ∈ 𝑠 ↔ (𝑥 ∈ 𝑊 ∧ (𝐺‘𝑥) ∈ 𝑠))) |
| 28 | 25, 27 | bitr4d 282 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ (◡𝐺 “ 𝑠) ↔ (𝐺‘𝑥) ∈ 𝑠)) |
| 29 | 21, 28 | anbi12d 632 |
. . . . . . . . 9
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → ((𝑥 ∈ (◡𝐹 “ 𝑟) ∧ 𝑥 ∈ (◡𝐺 “ 𝑠)) ↔ ((𝐹‘𝑥) ∈ 𝑟 ∧ (𝐺‘𝑥) ∈ 𝑠))) |
| 30 | | elin 3967 |
. . . . . . . . 9
⊢ (𝑥 ∈ ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠)) ↔ (𝑥 ∈ (◡𝐹 “ 𝑟) ∧ 𝑥 ∈ (◡𝐺 “ 𝑠))) |
| 31 | | opelxp 5721 |
. . . . . . . . 9
⊢
(〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝑟 × 𝑠) ↔ ((𝐹‘𝑥) ∈ 𝑟 ∧ (𝐺‘𝑥) ∈ 𝑠)) |
| 32 | 29, 30, 31 | 3bitr4g 314 |
. . . . . . . 8
⊢ ((((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠)) ↔ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝑟 × 𝑠))) |
| 33 | 32 | rabbi2dva 4226 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑊 ∩ ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠))) = {𝑥 ∈ 𝑊 ∣ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝑟 × 𝑠)}) |
| 34 | | inss1 4237 |
. . . . . . . . . 10
⊢ ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠)) ⊆ (◡𝐹 “ 𝑟) |
| 35 | | cnvimass 6100 |
. . . . . . . . . 10
⊢ (◡𝐹 “ 𝑟) ⊆ dom 𝐹 |
| 36 | 34, 35 | sstri 3993 |
. . . . . . . . 9
⊢ ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠)) ⊆ dom 𝐹 |
| 37 | 36, 14 | fssdm 6755 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠)) ⊆ 𝑊) |
| 38 | | sseqin2 4223 |
. . . . . . . 8
⊢ (((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠)) ⊆ 𝑊 ↔ (𝑊 ∩ ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠))) = ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠))) |
| 39 | 37, 38 | sylib 218 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑊 ∩ ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠))) = ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠))) |
| 40 | 33, 39 | eqtr3d 2779 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → {𝑥 ∈ 𝑊 ∣ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝑟 × 𝑠)} = ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠))) |
| 41 | 13, 40 | eqtrid 2789 |
. . . . 5
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (◡𝐻 “ (𝑟 × 𝑠)) = ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠))) |
| 42 | | cntop1 23248 |
. . . . . . . 8
⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑈 ∈ Top) |
| 43 | 42 | adantl 481 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ Top) |
| 44 | 43 | adantr 480 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑈 ∈ Top) |
| 45 | | cnima 23273 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝑟 ∈ 𝑅) → (◡𝐹 “ 𝑟) ∈ 𝑈) |
| 46 | 45 | ad2ant2r 747 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (◡𝐹 “ 𝑟) ∈ 𝑈) |
| 47 | | cnima 23273 |
. . . . . . 7
⊢ ((𝐺 ∈ (𝑈 Cn 𝑆) ∧ 𝑠 ∈ 𝑆) → (◡𝐺 “ 𝑠) ∈ 𝑈) |
| 48 | 47 | ad2ant2l 746 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (◡𝐺 “ 𝑠) ∈ 𝑈) |
| 49 | | inopn 22905 |
. . . . . 6
⊢ ((𝑈 ∈ Top ∧ (◡𝐹 “ 𝑟) ∈ 𝑈 ∧ (◡𝐺 “ 𝑠) ∈ 𝑈) → ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠)) ∈ 𝑈) |
| 50 | 44, 46, 48, 49 | syl3anc 1373 |
. . . . 5
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((◡𝐹 “ 𝑟) ∩ (◡𝐺 “ 𝑠)) ∈ 𝑈) |
| 51 | 41, 50 | eqeltrd 2841 |
. . . 4
⊢ (((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈) |
| 52 | 51 | ralrimivva 3202 |
. . 3
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈) |
| 53 | | vex 3484 |
. . . . . 6
⊢ 𝑟 ∈ V |
| 54 | | vex 3484 |
. . . . . 6
⊢ 𝑠 ∈ V |
| 55 | 53, 54 | xpex 7773 |
. . . . 5
⊢ (𝑟 × 𝑠) ∈ V |
| 56 | 55 | rgen2w 3066 |
. . . 4
⊢
∀𝑟 ∈
𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V |
| 57 | | eqid 2737 |
. . . . 5
⊢ (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) |
| 58 | | imaeq2 6074 |
. . . . . 6
⊢ (𝑧 = (𝑟 × 𝑠) → (◡𝐻 “ 𝑧) = (◡𝐻 “ (𝑟 × 𝑠))) |
| 59 | 58 | eleq1d 2826 |
. . . . 5
⊢ (𝑧 = (𝑟 × 𝑠) → ((◡𝐻 “ 𝑧) ∈ 𝑈 ↔ (◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)) |
| 60 | 57, 59 | ralrnmpo 7572 |
. . . 4
⊢
(∀𝑟 ∈
𝑅 ∀𝑠 ∈ 𝑆 (𝑟 × 𝑠) ∈ V → (∀𝑧 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(◡𝐻 “ 𝑧) ∈ 𝑈 ↔ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈)) |
| 61 | 56, 60 | ax-mp 5 |
. . 3
⊢
(∀𝑧 ∈
ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(◡𝐻 “ 𝑧) ∈ 𝑈 ↔ ∀𝑟 ∈ 𝑅 ∀𝑠 ∈ 𝑆 (◡𝐻 “ (𝑟 × 𝑠)) ∈ 𝑈) |
| 62 | 52, 61 | sylibr 234 |
. 2
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∀𝑧 ∈ ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(◡𝐻 “ 𝑧) ∈ 𝑈) |
| 63 | 1 | toptopon 22923 |
. . . 4
⊢ (𝑈 ∈ Top ↔ 𝑈 ∈ (TopOn‘𝑊)) |
| 64 | 43, 63 | sylib 218 |
. . 3
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝑈 ∈ (TopOn‘𝑊)) |
| 65 | | cntop2 23249 |
. . . 4
⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ Top) |
| 66 | | cntop2 23249 |
. . . 4
⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ Top) |
| 67 | | eqid 2737 |
. . . . 5
⊢ ran
(𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)) |
| 68 | 67 | txval 23572 |
. . . 4
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)))) |
| 69 | 65, 66, 68 | syl2an 596 |
. . 3
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠)))) |
| 70 | | toptopon2 22924 |
. . . . 5
⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅)) |
| 71 | 65, 70 | sylib 218 |
. . . 4
⊢ (𝐹 ∈ (𝑈 Cn 𝑅) → 𝑅 ∈ (TopOn‘∪ 𝑅)) |
| 72 | | toptopon2 22924 |
. . . . 5
⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆)) |
| 73 | 66, 72 | sylib 218 |
. . . 4
⊢ (𝐺 ∈ (𝑈 Cn 𝑆) → 𝑆 ∈ (TopOn‘∪ 𝑆)) |
| 74 | | txtopon 23599 |
. . . 4
⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅)
∧ 𝑆 ∈
(TopOn‘∪ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅
× ∪ 𝑆))) |
| 75 | 71, 73, 74 | syl2an 596 |
. . 3
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅
× ∪ 𝑆))) |
| 76 | 64, 69, 75 | tgcn 23260 |
. 2
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → (𝐻 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ (𝐻:𝑊⟶(∪ 𝑅 × ∪ 𝑆)
∧ ∀𝑧 ∈ ran
(𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ↦ (𝑟 × 𝑠))(◡𝐻 “ 𝑧) ∈ 𝑈))) |
| 77 | 12, 62, 76 | mpbir2and 713 |
1
⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻 ∈ (𝑈 Cn (𝑅 ×t 𝑆))) |