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Theorem fimaproj 8067
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 5421 . . . . 5 ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ V
2 fvproj.h . . . . . 6 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
3 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
4 vex 3449 . . . . . . . . . 10 𝑦 ∈ V
53, 4op1std 7931 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
65fveq2d 6846 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑧)) = (𝐹𝑥))
73, 4op2ndd 7932 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
87fveq2d 6846 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘(2nd𝑧)) = (𝐺𝑦))
96, 8opeq12d 4838 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ = ⟨(𝐹𝑥), (𝐺𝑦)⟩)
109mpompt 7470 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩) = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
112, 10eqtr4i 2767 . . . . 5 𝐻 = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩)
121, 11fnmpti 6644 . . . 4 𝐻 Fn (𝐴 × 𝐵)
13 fimaproj.x . . . . 5 (𝜑𝑋𝐴)
14 fimaproj.y . . . . 5 (𝜑𝑌𝐵)
15 xpss12 5648 . . . . 5 ((𝑋𝐴𝑌𝐵) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
1613, 14, 15syl2anc 584 . . . 4 (𝜑 → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
17 fvelimab 6914 . . . 4 ((𝐻 Fn (𝐴 × 𝐵) ∧ (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵)) → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐))
1812, 16, 17sylancr 587 . . 3 (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐))
19 simp-4r 782 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑎𝑋)
20 simplr 767 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑏𝑌)
21 opelxpi 5670 . . . . . . . 8 ((𝑎𝑋𝑏𝑌) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
2219, 20, 21syl2anc 584 . . . . . . 7 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
23 simpllr 774 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐹𝑎) = (1st𝑐))
24 simpr 485 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐺𝑏) = (2nd𝑐))
2523, 24opeq12d 4838 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → ⟨(𝐹𝑎), (𝐺𝑏)⟩ = ⟨(1st𝑐), (2nd𝑐)⟩)
2613ad5antr 732 . . . . . . . . . 10 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑋𝐴)
2726, 19sseldd 3945 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑎𝐴)
2814ad5antr 732 . . . . . . . . . 10 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑌𝐵)
2928, 20sseldd 3945 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑏𝐵)
302, 27, 29fvproj 8066 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = ⟨(𝐹𝑎), (𝐺𝑏)⟩)
31 1st2nd2 7960 . . . . . . . . 9 (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) → 𝑐 = ⟨(1st𝑐), (2nd𝑐)⟩)
3231ad5antlr 733 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑐 = ⟨(1st𝑐), (2nd𝑐)⟩)
3325, 30, 323eqtr4d 2786 . . . . . . 7 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐)
34 fveqeq2 6851 . . . . . . . 8 (𝑧 = ⟨𝑎, 𝑏⟩ → ((𝐻𝑧) = 𝑐 ↔ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐))
3534rspcev 3581 . . . . . . 7 ((⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌) ∧ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
3622, 33, 35syl2anc 584 . . . . . 6 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
37 fimaproj.g . . . . . . . . 9 (𝜑𝐺 Fn 𝐵)
3837ad3antrrr 728 . . . . . . . 8 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → 𝐺 Fn 𝐵)
39 fnfun 6602 . . . . . . . 8 (𝐺 Fn 𝐵 → Fun 𝐺)
4038, 39syl 17 . . . . . . 7 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → Fun 𝐺)
41 xp2nd 7954 . . . . . . . 8 (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) → (2nd𝑐) ∈ (𝐺𝑌))
4241ad3antlr 729 . . . . . . 7 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → (2nd𝑐) ∈ (𝐺𝑌))
43 fvelima 6908 . . . . . . 7 ((Fun 𝐺 ∧ (2nd𝑐) ∈ (𝐺𝑌)) → ∃𝑏𝑌 (𝐺𝑏) = (2nd𝑐))
4440, 42, 43syl2anc 584 . . . . . 6 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → ∃𝑏𝑌 (𝐺𝑏) = (2nd𝑐))
4536, 44r19.29a 3159 . . . . 5 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
46 fimaproj.f . . . . . . . 8 (𝜑𝐹 Fn 𝐴)
4746adantr 481 . . . . . . 7 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → 𝐹 Fn 𝐴)
48 fnfun 6602 . . . . . . 7 (𝐹 Fn 𝐴 → Fun 𝐹)
4947, 48syl 17 . . . . . 6 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → Fun 𝐹)
50 xp1st 7953 . . . . . . 7 (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) → (1st𝑐) ∈ (𝐹𝑋))
5150adantl 482 . . . . . 6 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → (1st𝑐) ∈ (𝐹𝑋))
52 fvelima 6908 . . . . . 6 ((Fun 𝐹 ∧ (1st𝑐) ∈ (𝐹𝑋)) → ∃𝑎𝑋 (𝐹𝑎) = (1st𝑐))
5349, 51, 52syl2anc 584 . . . . 5 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → ∃𝑎𝑋 (𝐹𝑎) = (1st𝑐))
5445, 53r19.29a 3159 . . . 4 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
55 simpr 485 . . . . . 6 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐻𝑧) = 𝑐)
5616ad2antrr 724 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
57 simplr 767 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑧 ∈ (𝑋 × 𝑌))
5856, 57sseldd 3945 . . . . . . . 8 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑧 ∈ (𝐴 × 𝐵))
5911fvmpt2 6959 . . . . . . . 8 ((𝑧 ∈ (𝐴 × 𝐵) ∧ ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ V) → (𝐻𝑧) = ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩)
6058, 1, 59sylancl 586 . . . . . . 7 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐻𝑧) = ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩)
6146ad2antrr 724 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝐹 Fn 𝐴)
6213ad2antrr 724 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑋𝐴)
63 xp1st 7953 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
6457, 63syl 17 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (1st𝑧) ∈ 𝑋)
65 fnfvima 7183 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑋𝐴 ∧ (1st𝑧) ∈ 𝑋) → (𝐹‘(1st𝑧)) ∈ (𝐹𝑋))
6661, 62, 64, 65syl3anc 1371 . . . . . . . 8 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐹‘(1st𝑧)) ∈ (𝐹𝑋))
6737ad2antrr 724 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝐺 Fn 𝐵)
6814ad2antrr 724 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑌𝐵)
69 xp2nd 7954 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
7057, 69syl 17 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (2nd𝑧) ∈ 𝑌)
71 fnfvima 7183 . . . . . . . . 9 ((𝐺 Fn 𝐵𝑌𝐵 ∧ (2nd𝑧) ∈ 𝑌) → (𝐺‘(2nd𝑧)) ∈ (𝐺𝑌))
7267, 68, 70, 71syl3anc 1371 . . . . . . . 8 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐺‘(2nd𝑧)) ∈ (𝐺𝑌))
73 opelxpi 5670 . . . . . . . 8 (((𝐹‘(1st𝑧)) ∈ (𝐹𝑋) ∧ (𝐺‘(2nd𝑧)) ∈ (𝐺𝑌)) → ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ ((𝐹𝑋) × (𝐺𝑌)))
7466, 72, 73syl2anc 584 . . . . . . 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 3155 . . . 4 ((𝜑 ∧ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐) → 𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)))
7854, 77impbida 799 . . 3 (𝜑 → (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐))
7918, 78bitr4d 281 . 2 (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ 𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))))
8079eqrdv 2734 1 (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹𝑋) × (𝐺𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  wss 3910  cop 4592  cmpt 5188   × cxp 5631  cima 5636  Fun wfun 6490   Fn wfn 6491  cfv 6496  cmpo 7359  1st c1st 7919  2nd c2nd 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922
This theorem is referenced by:  txomap  32415
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