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Theorem prfval 16760
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 16736 . . . 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 6158 . . . . . 6 (1st𝑓) ∈ V
54dmex 7046 . . . . 5 dom (1st𝑓) ∈ V
65a1i 11 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) ∈ V)
7 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → 𝑓 = 𝐹)
87fveq2d 6152 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (1st𝑓) = (1st𝐹))
98dmeqd 5286 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) = dom (1st𝐹))
10 prfval.b . . . . . . . 8 𝐵 = (Base‘𝐶)
11 eqid 2621 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
12 relfunc 16443 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
13 prfval.c . . . . . . . . 9 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
14 1st2ndbr 7162 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1512, 13, 14sylancr 694 . . . . . . . 8 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1610, 11, 15funcf1 16447 . . . . . . 7 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
17 fdm 6008 . . . . . . 7 ((1st𝐹):𝐵⟶(Base‘𝐷) → dom (1st𝐹) = 𝐵)
1816, 17syl 17 . . . . . 6 (𝜑 → dom (1st𝐹) = 𝐵)
1918adantr 481 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝐹) = 𝐵)
209, 19eqtrd 2655 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → dom (1st𝑓) = 𝐵)
21 simpr 477 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
22 simplrl 799 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑓 = 𝐹)
2322fveq2d 6152 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st𝑓) = (1st𝐹))
2423fveq1d 6150 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st𝑓)‘𝑥) = ((1st𝐹)‘𝑥))
25 simplrr 800 . . . . . . . . 9 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → 𝑔 = 𝐺)
2625fveq2d 6152 . . . . . . . 8 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (1st𝑔) = (1st𝐺))
2726fveq1d 6150 . . . . . . 7 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ((1st𝑔)‘𝑥) = ((1st𝐺)‘𝑥))
2824, 27opeq12d 4378 . . . . . 6 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
2921, 28mpteq12dv 4693 . . . . 5 (((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) → (𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩) = (𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
30 eqidd 2622 . . . . . . 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)‘)⟩))
3121, 21, 30mpt2eq123dv 6670 . . . . . 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)‘)⟩)))
3222ad2antrr 761 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑓 = 𝐹)
3332fveq2d 6152 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (2nd𝑓) = (2nd𝐹))
3433oveqd 6621 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(2nd𝑓)𝑦) = (𝑥(2nd𝐹)𝑦))
3534dmeqd 5286 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → dom (𝑥(2nd𝑓)𝑦) = dom (𝑥(2nd𝐹)𝑦))
36 prfval.h . . . . . . . . . . . 12 𝐻 = (Hom ‘𝐶)
37 eqid 2621 . . . . . . . . . . . 12 (Hom ‘𝐷) = (Hom ‘𝐷)
3815ad4antr 767 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
39 simplr 791 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑥𝐵)
40 simpr 477 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
4110, 36, 37, 38, 39, 40funcf2 16449 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(2nd𝐹)𝑦):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
42 fdm 6008 . . . . . . . . . . 11 ((𝑥(2nd𝐹)𝑦):(𝑥𝐻𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) → dom (𝑥(2nd𝐹)𝑦) = (𝑥𝐻𝑦))
4341, 42syl 17 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → dom (𝑥(2nd𝐹)𝑦) = (𝑥𝐻𝑦))
4435, 43eqtrd 2655 . . . . . . . . 9 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → dom (𝑥(2nd𝑓)𝑦) = (𝑥𝐻𝑦))
4534fveq1d 6150 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(2nd𝑓)𝑦)‘) = ((𝑥(2nd𝐹)𝑦)‘))
4625ad2antrr 761 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → 𝑔 = 𝐺)
4746fveq2d 6152 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (2nd𝑔) = (2nd𝐺))
4847oveqd 6621 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → (𝑥(2nd𝑔)𝑦) = (𝑥(2nd𝐺)𝑦))
4948fveq1d 6150 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ((𝑥(2nd𝑔)𝑦)‘) = ((𝑥(2nd𝐺)𝑦)‘))
5045, 49opeq12d 4378 . . . . . . . . 9 (((((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) ∧ 𝑏 = 𝐵) ∧ 𝑥𝐵) ∧ 𝑦𝐵) → ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩ = ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)
5144, 50mpteq12dv 4693 . . . . . . . 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)‘)⟩))
52513impa 1256 . . . . . . 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)‘)⟩))
5352mpt2eq3dva 6672 . . . . . 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)‘)⟩)))
5431, 53eqtrd 2655 . . . . 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)‘)⟩)))
5529, 54opeq12d 4378 . . . 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)‘)⟩))⟩)
566, 20, 55csbied2 3542 . . 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)‘)⟩))⟩)
57 elex 3198 . . . 4 (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 ∈ V)
5813, 57syl 17 . . 3 (𝜑𝐹 ∈ V)
59 prfval.d . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
60 elex 3198 . . . 4 (𝐺 ∈ (𝐶 Func 𝐸) → 𝐺 ∈ V)
6159, 60syl 17 . . 3 (𝜑𝐺 ∈ V)
62 opex 4893 . . . 4 ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ ∈ V
6362a1i 11 . . 3 (𝜑 → ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ ∈ V)
643, 56, 58, 61, 63ovmpt2d 6741 . 2 (𝜑 → (𝐹 ⟨,⟩F 𝐺) = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
651, 64syl5eq 2667 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 384   = wceq 1480  wcel 1987  Vcvv 3186  csb 3514  cop 4154   class class class wbr 4613  cmpt 4673  dom cdm 5074  Rel wrel 5079  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  1st c1st 7111  2nd c2nd 7112  Basecbs 15781  Hom chom 15873   Func cfunc 16435   ⟨,⟩F cprf 16732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-ixp 7853  df-func 16439  df-prf 16736
This theorem is referenced by:  prf1  16761  prf2fval  16762  prfcl  16764  prf1st  16765  prf2nd  16766  1st2ndprf  16767
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