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Theorem upxp 14249
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 5762 . . . 4 (𝐴𝐷 → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ V)
2 eueq 2923 . . . 4 ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ∈ V ↔ ∃! = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
31, 2sylib 122 . . 3 (𝐴𝐷 → ∃! = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
433ad2ant1 1020 . 2 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∃! = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
5 ffn 5384 . . . . . . . 8 (:𝐴⟶(𝐵 × 𝐶) → Fn 𝐴)
653ad2ant1 1020 . . . . . . 7 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → Fn 𝐴)
76adantl 277 . . . . . 6 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → Fn 𝐴)
8 ffvelcdm 5670 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
9 ffvelcdm 5670 . . . . . . . . . . . . 13 ((𝐺:𝐴𝐶𝑥𝐴) → (𝐺𝑥) ∈ 𝐶)
10 opelxpi 4676 . . . . . . . . . . . . 13 (((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐶) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
118, 9, 10syl2an 289 . . . . . . . . . . . 12 (((𝐹:𝐴𝐵𝑥𝐴) ∧ (𝐺:𝐴𝐶𝑥𝐴)) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
1211anandirs 593 . . . . . . . . . . 11 (((𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
1312ralrimiva 2563 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
14133adant1 1017 . . . . . . . . 9 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
15 eqid 2189 . . . . . . . . . 10 (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
1615fmpt 5687 . . . . . . . . 9 (∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶))
1714, 16sylib 122 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶))
1817ffnd 5385 . . . . . . 7 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴)
1918adantr 276 . . . . . 6 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴)
20 xpss 4752 . . . . . . . . . . 11 (𝐵 × 𝐶) ⊆ (V × V)
21 ffvelcdm 5670 . . . . . . . . . . 11 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (𝑧) ∈ (𝐵 × 𝐶))
2220, 21sselid 3168 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (𝑧) ∈ (V × V))
23223ad2antl1 1161 . . . . . . . . 9 (((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ∧ 𝑧𝐴) → (𝑧) ∈ (V × V))
2423adantll 476 . . . . . . . 8 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝑧) ∈ (V × V))
25 fveq1 5533 . . . . . . . . . . . 12 (𝐹 = (𝑃) → (𝐹𝑧) = ((𝑃)‘𝑧))
26 upxp.1 . . . . . . . . . . . . . 14 𝑃 = (1st ↾ (𝐵 × 𝐶))
2726coeq1i 4804 . . . . . . . . . . . . 13 (𝑃) = ((1st ↾ (𝐵 × 𝐶)) ∘ )
2827fveq1i 5535 . . . . . . . . . . . 12 ((𝑃)‘𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧)
2925, 28eqtrdi 2238 . . . . . . . . . . 11 (𝐹 = (𝑃) → (𝐹𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
30293ad2ant2 1021 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → (𝐹𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
3130ad2antlr 489 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝐹𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
32 simpr1 1005 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → :𝐴⟶(𝐵 × 𝐶))
33 fvco3 5608 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘(𝑧)))
3432, 33sylan 283 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ )‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘(𝑧)))
35213ad2antl1 1161 . . . . . . . . . . 11 (((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ∧ 𝑧𝐴) → (𝑧) ∈ (𝐵 × 𝐶))
3635adantll 476 . . . . . . . . . 10 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝑧) ∈ (𝐵 × 𝐶))
3736fvresd 5559 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → ((1st ↾ (𝐵 × 𝐶))‘(𝑧)) = (1st ‘(𝑧)))
3831, 34, 373eqtrrd 2227 . . . . . . . 8 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (1st ‘(𝑧)) = (𝐹𝑧))
39 fveq1 5533 . . . . . . . . . . . 12 (𝐺 = (𝑄) → (𝐺𝑧) = ((𝑄)‘𝑧))
40 upxp.2 . . . . . . . . . . . . . 14 𝑄 = (2nd ↾ (𝐵 × 𝐶))
4140coeq1i 4804 . . . . . . . . . . . . 13 (𝑄) = ((2nd ↾ (𝐵 × 𝐶)) ∘ )
4241fveq1i 5535 . . . . . . . . . . . 12 ((𝑄)‘𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧)
4339, 42eqtrdi 2238 . . . . . . . . . . 11 (𝐺 = (𝑄) → (𝐺𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
44433ad2ant3 1022 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → (𝐺𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
4544ad2antlr 489 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝐺𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧))
46 fvco3 5608 . . . . . . . . . 10 ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘(𝑧)))
4732, 46sylan 283 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ )‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘(𝑧)))
4836fvresd 5559 . . . . . . . . 9 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘(𝑧)) = (2nd ‘(𝑧)))
4945, 47, 483eqtrrd 2227 . . . . . . . 8 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (2nd ‘(𝑧)) = (𝐺𝑧))
50 eqopi 6198 . . . . . . . 8 (((𝑧) ∈ (V × V) ∧ ((1st ‘(𝑧)) = (𝐹𝑧) ∧ (2nd ‘(𝑧)) = (𝐺𝑧))) → (𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
5124, 38, 49, 50syl12anc 1247 . . . . . . 7 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
52 fveq2 5534 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
53 fveq2 5534 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐺𝑥) = (𝐺𝑧))
5452, 53opeq12d 3801 . . . . . . . 8 (𝑥 = 𝑧 → ⟨(𝐹𝑥), (𝐺𝑥)⟩ = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
55 simpr 110 . . . . . . . 8 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → 𝑧𝐴)
5651, 36eqeltrrd 2267 . . . . . . . 8 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
5715, 54, 55, 56fvmptd3 5630 . . . . . . 7 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
5851, 57eqtr4d 2225 . . . . . 6 ((((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) ∧ 𝑧𝐴) → (𝑧) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧))
597, 19, 58eqfnfvd 5637 . . . . 5 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))) → = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
6059ex 115 . . . 4 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) → = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
61 ffn 5384 . . . . . . . . 9 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
62613ad2ant2 1021 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐹 Fn 𝐴)
63 fo1st 6183 . . . . . . . . . . 11 1st :V–onto→V
64 fofn 5459 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
6563, 64ax-mp 5 . . . . . . . . . 10 1st Fn V
66 ssv 3192 . . . . . . . . . 10 (𝐵 × 𝐶) ⊆ V
67 fnssres 5348 . . . . . . . . . 10 ((1st Fn V ∧ (𝐵 × 𝐶) ⊆ V) → (1st ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶))
6865, 66, 67mp2an 426 . . . . . . . . 9 (1st ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶)
6917frnd 5394 . . . . . . . . 9 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ran (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ⊆ (𝐵 × 𝐶))
70 fnco 5343 . . . . . . . . 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)
7168, 18, 69, 70mp3an2i 1353 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝐴)
72 fvco3 5608 . . . . . . . . . 10 (((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
7317, 72sylan 283 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((1st ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
74 simpr 110 . . . . . . . . . . 11 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → 𝑧𝐴)
75 simpl2 1003 . . . . . . . . . . . . 13 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → 𝐹:𝐴𝐵)
7675, 74ffvelcdmd 5673 . . . . . . . . . . . 12 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
77 simpl3 1004 . . . . . . . . . . . . 13 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → 𝐺:𝐴𝐶)
7877, 74ffvelcdmd 5673 . . . . . . . . . . . 12 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (𝐺𝑧) ∈ 𝐶)
7976, 78opelxpd 4677 . . . . . . . . . . 11 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
8015, 54, 74, 79fvmptd3 5630 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧) = ⟨(𝐹𝑧), (𝐺𝑧)⟩)
8180fveq2d 5538 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((1st ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
82 ffvelcdm 5670 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
83 ffvelcdm 5670 . . . . . . . . . . . . . 14 ((𝐺:𝐴𝐶𝑧𝐴) → (𝐺𝑧) ∈ 𝐶)
84 opelxpi 4676 . . . . . . . . . . . . . 14 (((𝐹𝑧) ∈ 𝐵 ∧ (𝐺𝑧) ∈ 𝐶) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
8582, 83, 84syl2an 289 . . . . . . . . . . . . 13 (((𝐹:𝐴𝐵𝑧𝐴) ∧ (𝐺:𝐴𝐶𝑧𝐴)) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
8685anandirs 593 . . . . . . . . . . . 12 (((𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
87863adantl1 1155 . . . . . . . . . . 11 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ⟨(𝐹𝑧), (𝐺𝑧)⟩ ∈ (𝐵 × 𝐶))
8887fvresd 5559 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
89 op1stg 6176 . . . . . . . . . . 11 (((𝐹𝑧) ∈ 𝐵 ∧ (𝐺𝑧) ∈ 𝐶) → (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧))
9076, 78, 89syl2anc 411 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (1st ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧))
9188, 90eqtrd 2222 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((1st ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐹𝑧))
9273, 81, 913eqtrrd 2227 . . . . . . . 8 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (𝐹𝑧) = (((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
9362, 71, 92eqfnfvd 5637 . . . . . . 7 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐹 = ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
9426coeq1i 4804 . . . . . . 7 (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((1st ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
9593, 94eqtr4di 2240 . . . . . 6 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐹 = (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
96 ffn 5384 . . . . . . . . 9 (𝐺:𝐴𝐶𝐺 Fn 𝐴)
97963ad2ant3 1022 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐺 Fn 𝐴)
98 fo2nd 6184 . . . . . . . . . . 11 2nd :V–onto→V
99 fofn 5459 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
10098, 99ax-mp 5 . . . . . . . . . 10 2nd Fn V
101 fnssres 5348 . . . . . . . . . 10 ((2nd Fn V ∧ (𝐵 × 𝐶) ⊆ V) → (2nd ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶))
102100, 66, 101mp2an 426 . . . . . . . . 9 (2nd ↾ (𝐵 × 𝐶)) Fn (𝐵 × 𝐶)
103 fnco 5343 . . . . . . . . 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)
104102, 18, 69, 103mp3an2i 1353 . . . . . . . 8 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) Fn 𝐴)
105 fvco3 5608 . . . . . . . . . 10 (((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝑧𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
10617, 105sylan 283 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧) = ((2nd ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)))
10780fveq2d 5538 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑧)) = ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
10887fvresd 5559 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩))
109 op2ndg 6177 . . . . . . . . . . 11 (((𝐹𝑧) ∈ 𝐵 ∧ (𝐺𝑧) ∈ 𝐶) → (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧))
11076, 78, 109syl2anc 411 . . . . . . . . . 10 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (2nd ‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧))
111108, 110eqtrd 2222 . . . . . . . . 9 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → ((2nd ↾ (𝐵 × 𝐶))‘⟨(𝐹𝑧), (𝐺𝑧)⟩) = (𝐺𝑧))
112106, 107, 1113eqtrrd 2227 . . . . . . . 8 (((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) ∧ 𝑧𝐴) → (𝐺𝑧) = (((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))‘𝑧))
11397, 104, 112eqfnfvd 5637 . . . . . . 7 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐺 = ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
11440coeq1i 4804 . . . . . . 7 (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) = ((2nd ↾ (𝐵 × 𝐶)) ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
115113, 114eqtr4di 2240 . . . . . 6 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → 𝐺 = (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
11617, 95, 1153jca 1179 . . . . 5 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)) ∧ 𝐺 = (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
117 feq1 5367 . . . . . 6 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (:𝐴⟶(𝐵 × 𝐶) ↔ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩):𝐴⟶(𝐵 × 𝐶)))
118 coeq2 4803 . . . . . . 7 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑃) = (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
119118eqeq2d 2201 . . . . . 6 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐹 = (𝑃) ↔ 𝐹 = (𝑃 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
120 coeq2 4803 . . . . . . 7 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝑄) = (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
121120eqeq2d 2201 . . . . . 6 ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (𝐺 = (𝑄) ↔ 𝐺 = (𝑄 ∘ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))))
122117, 119, 1213anbi123d 1323 . . . . 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)⟩)))))
123116, 122syl5ibrcom 157 . . . 4 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ( = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) → (:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄))))
12460, 123impbid 129 . . 3 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ((:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
125124eubidv 2046 . 2 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → (∃!(:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)) ↔ ∃! = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)))
1264, 125mpbird 167 1 ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∃!(:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  ∃!weu 2038  wcel 2160  wral 2468  Vcvv 2752  wss 3144  cop 3610  cmpt 4079   × cxp 4642  ran crn 4645  cres 4646  ccom 4648   Fn wfn 5230  wf 5231  ontowfo 5233  cfv 5235  1st c1st 6164  2nd c2nd 6165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6166  df-2nd 6167
This theorem is referenced by:  uptx  14251  txcn  14252
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