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Theorem prfval 18079
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 18055 . . . 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 6852 . . . . . 6 (1st𝑓) ∈ V
54dmex 7844 . . . . 5 dom (1st𝑓) ∈ V
65a1i 11 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) ∈ V)
7 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → 𝑓 = 𝐹)
87fveq2d 6843 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (1st𝑓) = (1st𝐹))
98dmeqd 5859 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) = dom (1st𝐹))
10 prfval.b . . . . . . . 8 𝐵 = (Base‘𝐶)
11 eqid 2736 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
12 relfunc 17740 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
13 prfval.c . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
14 1st2ndbr 7970 . . . . . . . . 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 17744 . . . . . . 7 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
1716fdmd 6676 . . . . . 6 (𝜑 → dom (1st𝐹) = 𝐵)
1817adantr 481 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝐹) = 𝐵)
199, 18eqtrd 2776 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) = 𝐵)
20 simpr 485 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
21 simplrl 775 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑓 = 𝐹)
2221fveq2d 6843 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st𝑓) = (1st𝐹))
2322fveq1d 6841 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
24 simplrr 776 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑔 = 𝐺)
2524fveq2d 6843 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st𝑔) = (1st𝐺))
2625fveq1d 6841 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st𝑔)‘𝑥) = ((1st𝐺)‘𝑥))
2723, 26opeq12d 4836 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
2820, 27mpteq12dv 5194 . . . . 5 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩) = (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
29 eqidd 2737 . . . . . . 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 7428 . . . . . 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 724 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑓 = 𝐹)
3231fveq2d 6843 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (2nd𝑓) = (2nd𝐹))
3332oveqd 7370 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(2nd𝑓)𝑦) = (𝑥(2nd𝐹)𝑦))
3433dmeqd 5859 . . . . . . . . . 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 2736 . . . . . . . . . . . 12 (Hom ‘𝐷) = (Hom ‘𝐷)
3715ad4antr 730 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
38 simplr 767 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑥𝐵)
39 simpr 485 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
4010, 35, 36, 37, 38, 39funcf2 17746 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(2nd𝐹)𝑦):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
4140fdmd 6676 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → dom (𝑥(2nd𝐹)𝑦) = (𝑥𝐻𝑦))
4234, 41eqtrd 2776 . . . . . . . . 9 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → dom (𝑥(2nd𝑓)𝑦) = (𝑥𝐻𝑦))
4333fveq1d 6841 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(2nd𝑓)𝑦)‘) = ((𝑥(2nd𝐹)𝑦)‘))
4424ad2antrr 724 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑔 = 𝐺)
4544fveq2d 6843 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (2nd𝑔) = (2nd𝐺))
4645oveqd 7370 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(2nd𝑔)𝑦) = (𝑥(2nd𝐺)𝑦))
4746fveq1d 6841 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(2nd𝑔)𝑦)‘) = ((𝑥(2nd𝐺)𝑦)‘))
4843, 47opeq12d 4836 . . . . . . . . 9 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩ = ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)
4942, 48mpteq12dv 5194 . . . . . . . 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 7430 . . . . . 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 2776 . . . . 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 4836 . . . 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 3893 . . 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 3463 . . 3 (𝜑𝐹 ∈ V)
56 prfval.d . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
5756elexd 3463 . . 3 (𝜑𝐺 ∈ V)
58 opex 5419 . . . 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 7503 . 2 (𝜑 → (𝐹 ⟨,⟩F 𝐺) = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
611, 60eqtrid 2788 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 396   = wceq 1541  wcel 2106  Vcvv 3443  csb 3853  cop 4590   class class class wbr 5103  cmpt 5186  dom cdm 5631  Rel wrel 5636  cfv 6493  (class class class)co 7353  cmpo 7355  1st c1st 7915  2nd c2nd 7916  Basecbs 17075  Hom chom 17136   Func cfunc 17732   ⟨,⟩F cprf 18051
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-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
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 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-map 8763  df-ixp 8832  df-func 17736  df-prf 18055
This theorem is referenced by:  prf1  18080  prf2fval  18081  prfcl  18083  prf1st  18084  prf2nd  18085  1st2ndprf  18086
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