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Theorem fimaproj 8068
Description: Image of a cartesian product for a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)
Hypotheses
Ref Expression
fvproj.h 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
fimaproj.f (𝜑𝐹 Fn 𝐴)
fimaproj.g (𝜑𝐺 Fn 𝐵)
fimaproj.x (𝜑𝑋𝐴)
fimaproj.y (𝜑𝑌𝐵)
Assertion
Ref Expression
fimaproj (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹𝑋) × (𝐺𝑌)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem fimaproj
Dummy variables 𝑎 𝑏 𝑧 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5422 . . . . 5 ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ V
2 fvproj.h . . . . . 6 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
3 vex 3450 . . . . . . . . . 10 𝑥 ∈ V
4 vex 3450 . . . . . . . . . 10 𝑦 ∈ V
53, 4op1std 7932 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
65fveq2d 6847 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑧)) = (𝐹𝑥))
73, 4op2ndd 7933 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
87fveq2d 6847 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘(2nd𝑧)) = (𝐺𝑦))
96, 8opeq12d 4839 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ = ⟨(𝐹𝑥), (𝐺𝑦)⟩)
109mpompt 7471 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩) = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
112, 10eqtr4i 2768 . . . . 5 𝐻 = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩)
121, 11fnmpti 6645 . . . 4 𝐻 Fn (𝐴 × 𝐵)
13 fimaproj.x . . . . 5 (𝜑𝑋𝐴)
14 fimaproj.y . . . . 5 (𝜑𝑌𝐵)
15 xpss12 5649 . . . . 5 ((𝑋𝐴𝑌𝐵) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
1613, 14, 15syl2anc 585 . . . 4 (𝜑 → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
17 fvelimab 6915 . . . 4 ((𝐻 Fn (𝐴 × 𝐵) ∧ (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵)) → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐))
1812, 16, 17sylancr 588 . . 3 (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐))
19 simp-4r 783 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑎𝑋)
20 simplr 768 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑏𝑌)
21 opelxpi 5671 . . . . . . . 8 ((𝑎𝑋𝑏𝑌) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
2219, 20, 21syl2anc 585 . . . . . . 7 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
23 simpllr 775 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐹𝑎) = (1st𝑐))
24 simpr 486 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐺𝑏) = (2nd𝑐))
2523, 24opeq12d 4839 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → ⟨(𝐹𝑎), (𝐺𝑏)⟩ = ⟨(1st𝑐), (2nd𝑐)⟩)
2613ad5antr 733 . . . . . . . . . 10 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑋𝐴)
2726, 19sseldd 3946 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑎𝐴)
2814ad5antr 733 . . . . . . . . . 10 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑌𝐵)
2928, 20sseldd 3946 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑏𝐵)
302, 27, 29fvproj 8067 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = ⟨(𝐹𝑎), (𝐺𝑏)⟩)
31 1st2nd2 7961 . . . . . . . . 9 (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) → 𝑐 = ⟨(1st𝑐), (2nd𝑐)⟩)
3231ad5antlr 734 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑐 = ⟨(1st𝑐), (2nd𝑐)⟩)
3325, 30, 323eqtr4d 2787 . . . . . . 7 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐)
34 fveqeq2 6852 . . . . . . . 8 (𝑧 = ⟨𝑎, 𝑏⟩ → ((𝐻𝑧) = 𝑐 ↔ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐))
3534rspcev 3582 . . . . . . 7 ((⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌) ∧ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
3622, 33, 35syl2anc 585 . . . . . 6 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
37 fimaproj.g . . . . . . . . 9 (𝜑𝐺 Fn 𝐵)
3837ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → 𝐺 Fn 𝐵)
39 fnfun 6603 . . . . . . . 8 (𝐺 Fn 𝐵 → Fun 𝐺)
4038, 39syl 17 . . . . . . 7 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → Fun 𝐺)
41 xp2nd 7955 . . . . . . . 8 (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) → (2nd𝑐) ∈ (𝐺𝑌))
4241ad3antlr 730 . . . . . . 7 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → (2nd𝑐) ∈ (𝐺𝑌))
43 fvelima 6909 . . . . . . 7 ((Fun 𝐺 ∧ (2nd𝑐) ∈ (𝐺𝑌)) → ∃𝑏𝑌 (𝐺𝑏) = (2nd𝑐))
4440, 42, 43syl2anc 585 . . . . . 6 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → ∃𝑏𝑌 (𝐺𝑏) = (2nd𝑐))
4536, 44r19.29a 3160 . . . . 5 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
46 fimaproj.f . . . . . . . 8 (𝜑𝐹 Fn 𝐴)
4746adantr 482 . . . . . . 7 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → 𝐹 Fn 𝐴)
48 fnfun 6603 . . . . . . 7 (𝐹 Fn 𝐴 → Fun 𝐹)
4947, 48syl 17 . . . . . 6 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → Fun 𝐹)
50 xp1st 7954 . . . . . . 7 (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) → (1st𝑐) ∈ (𝐹𝑋))
5150adantl 483 . . . . . 6 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → (1st𝑐) ∈ (𝐹𝑋))
52 fvelima 6909 . . . . . 6 ((Fun 𝐹 ∧ (1st𝑐) ∈ (𝐹𝑋)) → ∃𝑎𝑋 (𝐹𝑎) = (1st𝑐))
5349, 51, 52syl2anc 585 . . . . 5 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → ∃𝑎𝑋 (𝐹𝑎) = (1st𝑐))
5445, 53r19.29a 3160 . . . 4 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
55 simpr 486 . . . . . 6 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐻𝑧) = 𝑐)
5616ad2antrr 725 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
57 simplr 768 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑧 ∈ (𝑋 × 𝑌))
5856, 57sseldd 3946 . . . . . . . 8 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑧 ∈ (𝐴 × 𝐵))
5911fvmpt2 6960 . . . . . . . 8 ((𝑧 ∈ (𝐴 × 𝐵) ∧ ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ V) → (𝐻𝑧) = ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩)
6058, 1, 59sylancl 587 . . . . . . 7 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐻𝑧) = ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩)
6146ad2antrr 725 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝐹 Fn 𝐴)
6213ad2antrr 725 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑋𝐴)
63 xp1st 7954 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
6457, 63syl 17 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (1st𝑧) ∈ 𝑋)
65 fnfvima 7184 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑋𝐴 ∧ (1st𝑧) ∈ 𝑋) → (𝐹‘(1st𝑧)) ∈ (𝐹𝑋))
6661, 62, 64, 65syl3anc 1372 . . . . . . . 8 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐹‘(1st𝑧)) ∈ (𝐹𝑋))
6737ad2antrr 725 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝐺 Fn 𝐵)
6814ad2antrr 725 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑌𝐵)
69 xp2nd 7955 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
7057, 69syl 17 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (2nd𝑧) ∈ 𝑌)
71 fnfvima 7184 . . . . . . . . 9 ((𝐺 Fn 𝐵𝑌𝐵 ∧ (2nd𝑧) ∈ 𝑌) → (𝐺‘(2nd𝑧)) ∈ (𝐺𝑌))
7267, 68, 70, 71syl3anc 1372 . . . . . . . 8 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐺‘(2nd𝑧)) ∈ (𝐺𝑌))
73 opelxpi 5671 . . . . . . . 8 (((𝐹‘(1st𝑧)) ∈ (𝐹𝑋) ∧ (𝐺‘(2nd𝑧)) ∈ (𝐺𝑌)) → ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ ((𝐹𝑋) × (𝐺𝑌)))
7466, 72, 73syl2anc 585 . . . . . . 7 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ ((𝐹𝑋) × (𝐺𝑌)))
7560, 74eqeltrd 2838 . . . . . 6 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐻𝑧) ∈ ((𝐹𝑋) × (𝐺𝑌)))
7655, 75eqeltrrd 2839 . . . . 5 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)))
7776r19.29an 3156 . . . 4 ((𝜑 ∧ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐) → 𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)))
7854, 77impbida 800 . . 3 (𝜑 → (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐))
7918, 78bitr4d 282 . 2 (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ 𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))))
8079eqrdv 2735 1 (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹𝑋) × (𝐺𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wrex 3074  Vcvv 3446  wss 3911  cop 4593  cmpt 5189   × cxp 5632  cima 5637  Fun wfun 6491   Fn wfn 6492  cfv 6497  cmpo 7360  1st c1st 7920  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923
This theorem is referenced by:  txomap  32418
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