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Theorem prfval 18244
Description: Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prfval.k 𝑃 = (𝐹 ⟨,⟩F 𝐺)
prfval.b 𝐵 = (Base‘𝐶)
prfval.h 𝐻 = (Hom ‘𝐶)
prfval.c (𝜑𝐹 ∈ (𝐶 Func 𝐷))
prfval.d (𝜑𝐺 ∈ (𝐶 Func 𝐸))
Assertion
Ref Expression
prfval (𝜑𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
Distinct variable groups:   𝑥,,𝑦,𝐵   𝑥,𝐶,𝑦   ,𝐹,𝑥,𝑦   𝜑,,𝑥,𝑦   𝑥,𝐷,𝑦   ,𝐺,𝑥,𝑦   ,𝐻,𝑥,𝑦
Allowed substitution hints:   𝐶()   𝐷()   𝑃(𝑥,𝑦,)   𝐸(𝑥,𝑦,)

Proof of Theorem prfval
Dummy variables 𝑓 𝑏 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfval.k . 2 𝑃 = (𝐹 ⟨,⟩F 𝐺)
2 df-prf 18220 . . . 4 ⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ dom (1st𝑓) / 𝑏⟨(𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩), (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩))⟩)
32a1i 11 . . 3 (𝜑 → ⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ dom (1st𝑓) / 𝑏⟨(𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩), (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩))⟩))
4 fvex 6919 . . . . . 6 (1st𝑓) ∈ V
54dmex 7931 . . . . 5 dom (1st𝑓) ∈ V
65a1i 11 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) ∈ V)
7 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → 𝑓 = 𝐹)
87fveq2d 6910 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (1st𝑓) = (1st𝐹))
98dmeqd 5916 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) = dom (1st𝐹))
10 prfval.b . . . . . . . 8 𝐵 = (Base‘𝐶)
11 eqid 2737 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
12 relfunc 17907 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
13 prfval.c . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
14 1st2ndbr 8067 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1512, 13, 14sylancr 587 . . . . . . . 8 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1610, 11, 15funcf1 17911 . . . . . . 7 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
1716fdmd 6746 . . . . . 6 (𝜑 → dom (1st𝐹) = 𝐵)
1817adantr 480 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝐹) = 𝐵)
199, 18eqtrd 2777 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) = 𝐵)
20 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
21 simplrl 777 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑓 = 𝐹)
2221fveq2d 6910 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st𝑓) = (1st𝐹))
2322fveq1d 6908 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
24 simplrr 778 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑔 = 𝐺)
2524fveq2d 6910 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st𝑔) = (1st𝐺))
2625fveq1d 6908 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st𝑔)‘𝑥) = ((1st𝐺)‘𝑥))
2723, 26opeq12d 4881 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
2820, 27mpteq12dv 5233 . . . . 5 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩) = (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
29 eqidd 2738 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩) = ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩))
3020, 20, 29mpoeq123dv 7508 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩)) = (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩)))
3121ad2antrr 726 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑓 = 𝐹)
3231fveq2d 6910 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (2nd𝑓) = (2nd𝐹))
3332oveqd 7448 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(2nd𝑓)𝑦) = (𝑥(2nd𝐹)𝑦))
3433dmeqd 5916 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → dom (𝑥(2nd𝑓)𝑦) = dom (𝑥(2nd𝐹)𝑦))
35 prfval.h . . . . . . . . . . . 12 𝐻 = (Hom ‘𝐶)
36 eqid 2737 . . . . . . . . . . . 12 (Hom ‘𝐷) = (Hom ‘𝐷)
3715ad4antr 732 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
38 simplr 769 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑥𝐵)
39 simpr 484 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
4010, 35, 36, 37, 38, 39funcf2 17913 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(2nd𝐹)𝑦):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
4140fdmd 6746 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → dom (𝑥(2nd𝐹)𝑦) = (𝑥𝐻𝑦))
4234, 41eqtrd 2777 . . . . . . . . 9 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → dom (𝑥(2nd𝑓)𝑦) = (𝑥𝐻𝑦))
4333fveq1d 6908 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(2nd𝑓)𝑦)‘) = ((𝑥(2nd𝐹)𝑦)‘))
4424ad2antrr 726 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑔 = 𝐺)
4544fveq2d 6910 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (2nd𝑔) = (2nd𝐺))
4645oveqd 7448 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(2nd𝑔)𝑦) = (𝑥(2nd𝐺)𝑦))
4746fveq1d 6908 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(2nd𝑔)𝑦)‘) = ((𝑥(2nd𝐺)𝑦)‘))
4843, 47opeq12d 4881 . . . . . . . . 9 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩ = ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)
4942, 48mpteq12dv 5233 . . . . . . . 8 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩) = ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
50493impa 1110 . . . . . . 7 ((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵𝑦𝐵) → ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩) = ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
5150mpoeq3dva 7510 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩)) = (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
5230, 51eqtrd 2777 . . . . 5 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩)) = (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
5328, 52opeq12d 4881 . . . 4 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ⟨(𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩), (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩))⟩ = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
546, 19, 53csbied2 3936 . . 3 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) / 𝑏⟨(𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩), (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩))⟩ = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
5513elexd 3504 . . 3 (𝜑𝐹 ∈ V)
56 prfval.d . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
5756elexd 3504 . . 3 (𝜑𝐺 ∈ V)
58 opex 5469 . . . 4 ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ ∈ V
5958a1i 11 . . 3 (𝜑 → ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ ∈ V)
603, 54, 55, 57, 59ovmpod 7585 . 2 (𝜑 → (𝐹 ⟨,⟩F 𝐺) = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
611, 60eqtrid 2789 1 (𝜑𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  csb 3899  cop 4632   class class class wbr 5143  cmpt 5225  dom cdm 5685  Rel wrel 5690  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308   Func cfunc 17899   ⟨,⟩F cprf 18216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ixp 8938  df-func 17903  df-prf 18220
This theorem is referenced by:  prf1  18245  prf2fval  18246  prfcl  18248  prf1st  18249  prf2nd  18250  1st2ndprf  18251
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