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Theorem upxp 21336
Description: Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
upxp.1 𝑃 = (1st ↾ (𝐵 × 𝐶))
upxp.2 𝑄 = (2nd ↾ (𝐵 × 𝐶))
Assertion
Ref Expression
upxp ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∃!(:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
Distinct variable groups:   𝐴,   𝐵,   𝐶,   ,𝐹   ,𝐺   𝐷,
Allowed substitution hints:   𝑃()   𝑄()

Proof of Theorem upxp
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mptexg 6438 . . . 4 (𝐴𝐷 → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ V)
2 eueq 3360 . . . 4 ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ V ↔ ∃! = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
31, 2sylib 208 . . 3 (𝐴𝐷 → ∃! = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
433ad2ant1 1080 . 2 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∃! = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
5 ffn 6002 . . . . . . . 8 (:𝐴⟶(𝐵 × 𝐶) → Fn 𝐴)
653ad2ant1 1080 . . . . . . 7 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → Fn 𝐴)
76adantl 482 . . . . . 6 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → Fn 𝐴)
8 ffvelrn 6313 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
9 ffvelrn 6313 . . . . . . . . . . . . 13 ((𝐺:𝐴𝐶𝑥𝐴) → (𝐺𝑥) ∈ 𝐶)
10 opelxpi 5108 . . . . . . . . . . . . 13 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐶) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
118, 9, 10syl2an 494 . . . . . . . . . . . 12 (((𝐹:𝐴𝐵𝑥𝐴) ∧ (𝐺:𝐴𝐶𝑥𝐴)) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
1211anandirs 873 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
1312ralrimiva 2960 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
14133adant1 1077 . . . . . . . . 9 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
15 eqid 2621 . . . . . . . . . 10 (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
1615fmpt 6337 . . . . . . . . 9 (∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶))
1714, 16sylib 208 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶))
18 ffn 6002 . . . . . . . 8 ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶) → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴)
1917, 18syl 17 . . . . . . 7 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴)
2019adantr 481 . . . . . 6 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴)
21 xpss 5187 . . . . . . . . . . 11 (𝐵 × 𝐶) ⊆ (V × V)
22 ffvelrn 6313 . . . . . . . . . . 11 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (𝑧) ∈ (𝐵 × 𝐶))
2321, 22sseldi 3581 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (𝑧) ∈ (V × V))
24233ad2antl1 1221 . . . . . . . . 9 (((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ∧ 𝑧𝐴) → (𝑧) ∈ (V × V))
2524adantll 749 . . . . . . . 8 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝑧) ∈ (V × V))
26 fveq1 6147 . . . . . . . . . . . 12 (𝐹 = (𝑃) → (𝐹𝑧) = ((𝑃)‘𝑧))
27 upxp.1 . . . . . . . . . . . . . 14 𝑃 = (1st ↾ (𝐵 × 𝐶))
2827coeq1i 5241 . . . . . . . . . . . . 13 (𝑃) = ((1st ↾ (𝐵 × 𝐶)) ∘ )
2928fveq1i 6149 . . . . . . . . . . . 12 ((𝑃)‘𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧)
3026, 29syl6eq 2671 . . . . . . . . . . 11 (𝐹 = (𝑃) → (𝐹𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
31303ad2ant2 1081 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → (𝐹𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
3231ad2antlr 762 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝐹𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
33 simpr1 1065 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → :𝐴⟶(𝐵 × 𝐶))
34 fvco3 6232 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘(𝑧)))
3533, 34sylan 488 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘(𝑧)))
36223ad2antl1 1221 . . . . . . . . . . 11 (((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ∧ 𝑧𝐴) → (𝑧) ∈ (𝐵 × 𝐶))
3736adantll 749 . . . . . . . . . 10 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝑧) ∈ (𝐵 × 𝐶))
38 fvres 6164 . . . . . . . . . 10 ((𝑧) ∈ (𝐵 × 𝐶) → ((1st ↾ (𝐵 × 𝐶))‘(𝑧)) = (1st ‘(𝑧)))
3937, 38syl 17 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → ((1st ↾ (𝐵 × 𝐶))‘(𝑧)) = (1st ‘(𝑧)))
4032, 35, 393eqtrrd 2660 . . . . . . . 8 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (1st ‘(𝑧)) = (𝐹𝑧))
41 fveq1 6147 . . . . . . . . . . . 12 (𝐺 = (𝑄) → (𝐺𝑧) = ((𝑄)‘𝑧))
42 upxp.2 . . . . . . . . . . . . . 14 𝑄 = (2nd ↾ (𝐵 × 𝐶))
4342coeq1i 5241 . . . . . . . . . . . . 13 (𝑄) = ((2nd ↾ (𝐵 × 𝐶)) ∘ )
4443fveq1i 6149 . . . . . . . . . . . 12 ((𝑄)‘𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧)
4541, 44syl6eq 2671 . . . . . . . . . . 11 (𝐺 = (𝑄) → (𝐺𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
46453ad2ant3 1082 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → (𝐺𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
4746ad2antlr 762 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝐺𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
48 fvco3 6232 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘(𝑧)))
4933, 48sylan 488 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘(𝑧)))
50 fvres 6164 . . . . . . . . . 10 ((𝑧) ∈ (𝐵 × 𝐶) → ((2nd ↾ (𝐵 × 𝐶))‘(𝑧)) = (2nd ‘(𝑧)))
5137, 50syl 17 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘(𝑧)) = (2nd ‘(𝑧)))
5247, 49, 513eqtrrd 2660 . . . . . . . 8 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (2nd ‘(𝑧)) = (𝐺𝑧))
53 eqopi 7147 . . . . . . . 8 (((𝑧) ∈ (V × V) ∧ ((1st ‘(𝑧)) = (𝐹𝑧) ∧ (2nd ‘(𝑧)) = (𝐺𝑧))) → (𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
5425, 40, 52, 53syl12anc 1321 . . . . . . 7 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
55 fveq2 6148 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
56 fveq2 6148 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
5755, 56opeq12d 4378 . . . . . . . . 9 (𝑥 = 𝑧 → ⟨(𝐹𝑥), (𝐺𝑥)⟩ = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
58 opex 4893 . . . . . . . . 9 ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ V
5957, 15, 58fvmpt 6239 . . . . . . . 8 (𝑧𝐴 → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
6059adantl 482 . . . . . . 7 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
6154, 60eqtr4d 2658 . . . . . 6 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝑧) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧))
627, 20, 61eqfnfvd 6270 . . . . 5 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
6362ex 450 . . . 4 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
64 ffn 6002 . . . . . . . . 9 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
65643ad2ant2 1081 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐹 Fn 𝐴)
66 fo1st 7133 . . . . . . . . . . . 12 1st :V–onto→V
67 fofn 6074 . . . . . . . . . . . 12 (1st :V–onto→V → 1st Fn V)
6866, 67ax-mp 5 . . . . . . . . . . 11 1st Fn V
69 ssv 3604 . . . . . . . . . . 11 (𝐵 × 𝐶) ⊆ V
70 fnssres 5962 . . . . . . . . . . 11 ((1st Fn V ∧ (𝐵 × 𝐶) ⊆ V) → (1st ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶))
7168, 69, 70mp2an 707 . . . . . . . . . 10 (1st ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶)
7271a1i 11 . . . . . . . . 9 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → (1st ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶))
73 frn 6010 . . . . . . . . . 10 ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶) → ran (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝐵 × 𝐶))
7417, 73syl 17 . . . . . . . . 9 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ran (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝐵 × 𝐶))
75 fnco 5957 . . . . . . . . 9 (((1st ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶) ∧ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴 ∧ ran (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝐵 × 𝐶)) → ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝐴)
7672, 19, 74, 75syl3anc 1323 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝐴)
77 fvco3 6232 . . . . . . . . . 10 (((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
7817, 77sylan 488 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
7959adantl 482 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
8079fveq2d 6152 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((1st ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
81 ffvelrn 6313 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
82 ffvelrn 6313 . . . . . . . . . . . . . 14 ((𝐺:𝐴𝐶𝑧𝐴) → (𝐺𝑧) ∈ 𝐶)
83 opelxpi 5108 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ 𝐵 ∧ (𝐺𝑧) ∈ 𝐶) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
8481, 82, 83syl2an 494 . . . . . . . . . . . . 13 (((𝐹:𝐴𝐵𝑧𝐴) ∧ (𝐺:𝐴𝐶𝑧𝐴)) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
8584anandirs 873 . . . . . . . . . . . 12 (((𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
86853adantl1 1215 . . . . . . . . . . 11 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
87 fvres 6164 . . . . . . . . . . 11 (⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶) → ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
8886, 87syl 17 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
89 fvex 6158 . . . . . . . . . . 11 (𝐹𝑧) ∈ V
90 fvex 6158 . . . . . . . . . . 11 (𝐺𝑧) ∈ V
9189, 90op1st 7121 . . . . . . . . . 10 (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧)
9288, 91syl6eq 2671 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧))
9378, 80, 923eqtrrd 2660 . . . . . . . 8 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (𝐹𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
9465, 76, 93eqfnfvd 6270 . . . . . . 7 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐹 = ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
9527coeq1i 5241 . . . . . . 7 (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
9694, 95syl6eqr 2673 . . . . . 6 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐹 = (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
97 ffn 6002 . . . . . . . . 9 (𝐺:𝐴𝐶𝐺 Fn 𝐴)
98973ad2ant3 1082 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐺 Fn 𝐴)
99 fo2nd 7134 . . . . . . . . . . . 12 2nd :V–onto→V
100 fofn 6074 . . . . . . . . . . . 12 (2nd :V–onto→V → 2nd Fn V)
10199, 100ax-mp 5 . . . . . . . . . . 11 2nd Fn V
102 fnssres 5962 . . . . . . . . . . 11 ((2nd Fn V ∧ (𝐵 × 𝐶) ⊆ V) → (2nd ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶))
103101, 69, 102mp2an 707 . . . . . . . . . 10 (2nd ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶)
104103a1i 11 . . . . . . . . 9 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → (2nd ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶))
105 fnco 5957 . . . . . . . . 9 (((2nd ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶) ∧ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴 ∧ ran (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝐵 × 𝐶)) → ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝐴)
106104, 19, 74, 105syl3anc 1323 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝐴)
107 fvco3 6232 . . . . . . . . . 10 (((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
10817, 107sylan 488 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
10979fveq2d 6152 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
110 fvres 6164 . . . . . . . . . . 11 (⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶) → ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
11186, 110syl 17 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
11289, 90op2nd 7122 . . . . . . . . . 10 (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧)
113111, 112syl6eq 2671 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧))
114108, 109, 1133eqtrrd 2660 . . . . . . . 8 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (𝐺𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
11598, 106, 114eqfnfvd 6270 . . . . . . 7 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐺 = ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
11642coeq1i 5241 . . . . . . 7 (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
117115, 116syl6eqr 2673 . . . . . 6 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐺 = (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
11817, 96, 1173jca 1240 . . . . 5 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
119 feq1 5983 . . . . . 6 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (:𝐴⟶(𝐵 × 𝐶) ↔ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶)))
120 coeq2 5240 . . . . . . 7 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑃) = (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
121120eqeq2d 2631 . . . . . 6 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐹 = (𝑃) ↔ 𝐹 = (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
122 coeq2 5240 . . . . . . 7 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑄) = (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
123122eqeq2d 2631 . . . . . 6 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐺 = (𝑄) ↔ 𝐺 = (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
124119, 121, 1233anbi123d 1396 . . . . 5 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))))
125118, 124syl5ibrcom 237 . . . 4 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
12663, 125impbid 202 . . 3 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
127126eubidv 2489 . 2 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → (∃!(:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ ∃! = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
1284, 127mpbird 247 1 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∃!(:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  ∃!weu 2469  wral 2907  Vcvv 3186  wss 3555  cop 4154  cmpt 4673   × cxp 5072  ran crn 5075  cres 5076  ccom 5078   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  1st c1st 7111  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1st 7113  df-2nd 7114
This theorem is referenced by:  uptx  21338  txcn  21339
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