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Theorem fimaproj 30408
Description: Image of a cartesian product for a function on pairs, given two projections 𝐹 and 𝐺. (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 5121 . . . . 5 ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ V
2 fvproj.h . . . . . 6 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
3 vex 3386 . . . . . . . . . 10 𝑥 ∈ V
4 vex 3386 . . . . . . . . . 10 𝑦 ∈ V
53, 4op1std 7409 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
65fveq2d 6413 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑧)) = (𝐹𝑥))
73, 4op2ndd 7410 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
87fveq2d 6413 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘(2nd𝑧)) = (𝐺𝑦))
96, 8opeq12d 4599 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ = ⟨(𝐹𝑥), (𝐺𝑦)⟩)
109mpt2mpt 6984 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩) = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
112, 10eqtr4i 2822 . . . . 5 𝐻 = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩)
121, 11fnmpti 6231 . . . 4 𝐻 Fn (𝐴 × 𝐵)
13 fimaproj.x . . . . 5 (𝜑𝑋𝐴)
14 fimaproj.y . . . . 5 (𝜑𝑌𝐵)
15 xpss12 5325 . . . . 5 ((𝑋𝐴𝑌𝐵) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
1613, 14, 15syl2anc 580 . . . 4 (𝜑 → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
17 fvelimab 6476 . . . 4 ((𝐻 Fn (𝐴 × 𝐵) ∧ (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵)) → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐))
1812, 16, 17sylancr 582 . . 3 (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐))
19 simp-4r 804 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑎𝑋)
20 simplr 786 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑏𝑌)
21 opelxpi 5347 . . . . . . . 8 ((𝑎𝑋𝑏𝑌) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
2219, 20, 21syl2anc 580 . . . . . . 7 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
23 simpllr 794 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐹𝑎) = (1st𝑐))
24 simpr 478 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐺𝑏) = (2nd𝑐))
2523, 24opeq12d 4599 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → ⟨(𝐹𝑎), (𝐺𝑏)⟩ = ⟨(1st𝑐), (2nd𝑐)⟩)
2613ad5antr 729 . . . . . . . . . 10 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑋𝐴)
2726, 19sseldd 3797 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑎𝐴)
2814ad5antr 729 . . . . . . . . . 10 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑌𝐵)
2928, 20sseldd 3797 . . . . . . . . 9 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑏𝐵)
302, 27, 29fvproj 30407 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = ⟨(𝐹𝑎), (𝐺𝑏)⟩)
31 1st2nd2 7438 . . . . . . . . 9 (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) → 𝑐 = ⟨(1st𝑐), (2nd𝑐)⟩)
3231ad5antlr 731 . . . . . . . 8 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → 𝑐 = ⟨(1st𝑐), (2nd𝑐)⟩)
3325, 30, 323eqtr4d 2841 . . . . . . 7 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐)
34 fveqeq2 6418 . . . . . . . 8 (𝑧 = ⟨𝑎, 𝑏⟩ → ((𝐻𝑧) = 𝑐 ↔ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐))
3534rspcev 3495 . . . . . . 7 ((⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌) ∧ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
3622, 33, 35syl2anc 580 . . . . . 6 ((((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) ∧ 𝑏𝑌) ∧ (𝐺𝑏) = (2nd𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
37 fimaproj.g . . . . . . . . 9 (𝜑𝐺 Fn 𝐵)
3837ad3antrrr 722 . . . . . . . 8 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → 𝐺 Fn 𝐵)
39 fnfun 6197 . . . . . . . 8 (𝐺 Fn 𝐵 → Fun 𝐺)
4038, 39syl 17 . . . . . . 7 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → Fun 𝐺)
41 xp2nd 7432 . . . . . . . 8 (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) → (2nd𝑐) ∈ (𝐺𝑌))
4241ad3antlr 723 . . . . . . 7 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → (2nd𝑐) ∈ (𝐺𝑌))
43 fvelima 6471 . . . . . . 7 ((Fun 𝐺 ∧ (2nd𝑐) ∈ (𝐺𝑌)) → ∃𝑏𝑌 (𝐺𝑏) = (2nd𝑐))
4440, 42, 43syl2anc 580 . . . . . 6 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → ∃𝑏𝑌 (𝐺𝑏) = (2nd𝑐))
4536, 44r19.29a 3257 . . . . 5 ((((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = (1st𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
46 fimaproj.f . . . . . . . 8 (𝜑𝐹 Fn 𝐴)
4746adantr 473 . . . . . . 7 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → 𝐹 Fn 𝐴)
48 fnfun 6197 . . . . . . 7 (𝐹 Fn 𝐴 → Fun 𝐹)
4947, 48syl 17 . . . . . 6 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → Fun 𝐹)
50 xp1st 7431 . . . . . . 7 (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) → (1st𝑐) ∈ (𝐹𝑋))
5150adantl 474 . . . . . 6 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → (1st𝑐) ∈ (𝐹𝑋))
52 fvelima 6471 . . . . . 6 ((Fun 𝐹 ∧ (1st𝑐) ∈ (𝐹𝑋)) → ∃𝑎𝑋 (𝐹𝑎) = (1st𝑐))
5349, 51, 52syl2anc 580 . . . . 5 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → ∃𝑎𝑋 (𝐹𝑎) = (1st𝑐))
5445, 53r19.29a 3257 . . . 4 ((𝜑𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐)
55 simpr 478 . . . . . 6 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐻𝑧) = 𝑐)
5616ad2antrr 718 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
57 simplr 786 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑧 ∈ (𝑋 × 𝑌))
5856, 57sseldd 3797 . . . . . . . 8 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑧 ∈ (𝐴 × 𝐵))
5911fvmpt2 6514 . . . . . . . 8 ((𝑧 ∈ (𝐴 × 𝐵) ∧ ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ V) → (𝐻𝑧) = ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩)
6058, 1, 59sylancl 581 . . . . . . 7 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐻𝑧) = ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩)
6146ad2antrr 718 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝐹 Fn 𝐴)
6213ad2antrr 718 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑋𝐴)
63 xp1st 7431 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
6457, 63syl 17 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (1st𝑧) ∈ 𝑋)
65 fnfvima 6723 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑋𝐴 ∧ (1st𝑧) ∈ 𝑋) → (𝐹‘(1st𝑧)) ∈ (𝐹𝑋))
6661, 62, 64, 65syl3anc 1491 . . . . . . . 8 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐹‘(1st𝑧)) ∈ (𝐹𝑋))
6737ad2antrr 718 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝐺 Fn 𝐵)
6814ad2antrr 718 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑌𝐵)
69 xp2nd 7432 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
7057, 69syl 17 . . . . . . . . 9 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (2nd𝑧) ∈ 𝑌)
71 fnfvima 6723 . . . . . . . . 9 ((𝐺 Fn 𝐵𝑌𝐵 ∧ (2nd𝑧) ∈ 𝑌) → (𝐺‘(2nd𝑧)) ∈ (𝐺𝑌))
7267, 68, 70, 71syl3anc 1491 . . . . . . . 8 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐺‘(2nd𝑧)) ∈ (𝐺𝑌))
73 opelxpi 5347 . . . . . . . 8 (((𝐹‘(1st𝑧)) ∈ (𝐹𝑋) ∧ (𝐺‘(2nd𝑧)) ∈ (𝐺𝑌)) → ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ ((𝐹𝑋) × (𝐺𝑌)))
7466, 72, 73syl2anc 580 . . . . . . 7 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → ⟨(𝐹‘(1st𝑧)), (𝐺‘(2nd𝑧))⟩ ∈ ((𝐹𝑋) × (𝐺𝑌)))
7560, 74eqeltrd 2876 . . . . . 6 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → (𝐻𝑧) ∈ ((𝐹𝑋) × (𝐺𝑌)))
7655, 75eqeltrrd 2877 . . . . 5 (((𝜑𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻𝑧) = 𝑐) → 𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)))
7776r19.29an 3256 . . . 4 ((𝜑 ∧ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐) → 𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)))
7854, 77impbida 836 . . 3 (𝜑 → (𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻𝑧) = 𝑐))
7918, 78bitr4d 274 . 2 (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ 𝑐 ∈ ((𝐹𝑋) × (𝐺𝑌))))
8079eqrdv 2795 1 (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹𝑋) × (𝐺𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wrex 3088  Vcvv 3383  wss 3767  cop 4372  cmpt 4920   × cxp 5308  cima 5313  Fun wfun 6093   Fn wfn 6094  cfv 6099  cmpt2 6878  1st c1st 7397  2nd c2nd 7398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400
This theorem is referenced by:  txomap  30409
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