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Theorem pw2f1olem 8180
Description: Lemma for pw2f1o 8181. (Contributed by Mario Carneiro, 6-Oct-2014.)
Hypotheses
Ref Expression
pw2f1o.1 (𝜑𝐴𝑉)
pw2f1o.2 (𝜑𝐵𝑊)
pw2f1o.3 (𝜑𝐶𝑊)
pw2f1o.4 (𝜑𝐵𝐶)
Assertion
Ref Expression
pw2f1olem (𝜑 → ((𝑆 ∈ 𝒫 𝐴𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵))) ↔ (𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑆 = (𝐺 “ {𝐶}))))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐶   𝑧,𝑆
Allowed substitution hints:   𝜑(𝑧)   𝐺(𝑧)   𝑉(𝑧)   𝑊(𝑧)

Proof of Theorem pw2f1olem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pw2f1o.3 . . . . . . . . . 10 (𝜑𝐶𝑊)
2 prid2g 4403 . . . . . . . . . 10 (𝐶𝑊𝐶 ∈ {𝐵, 𝐶})
31, 2syl 17 . . . . . . . . 9 (𝜑𝐶 ∈ {𝐵, 𝐶})
4 pw2f1o.2 . . . . . . . . . 10 (𝜑𝐵𝑊)
5 prid1g 4402 . . . . . . . . . 10 (𝐵𝑊𝐵 ∈ {𝐵, 𝐶})
64, 5syl 17 . . . . . . . . 9 (𝜑𝐵 ∈ {𝐵, 𝐶})
73, 6ifcld 4239 . . . . . . . 8 (𝜑 → if(𝑦𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
87adantr 472 . . . . . . 7 ((𝜑𝑦𝐴) → if(𝑦𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
9 eqid 2724 . . . . . . 7 (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))
108, 9fmptd 6500 . . . . . 6 (𝜑 → (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
1110adantr 472 . . . . 5 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
12 simprr 813 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
1312feq1d 6143 . . . . 5 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ↔ (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶}))
1411, 13mpbird 247 . . . 4 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺:𝐴⟶{𝐵, 𝐶})
15 iftrue 4200 . . . . . . . . 9 (𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) = 𝐶)
16 pw2f1o.4 . . . . . . . . . . . 12 (𝜑𝐵𝐶)
1716ad2antrr 764 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → 𝐵𝐶)
18 iffalse 4203 . . . . . . . . . . . 12 𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) = 𝐵)
1918neeq1d 2955 . . . . . . . . . . 11 𝑥𝑆 → (if(𝑥𝑆, 𝐶, 𝐵) ≠ 𝐶𝐵𝐶))
2017, 19syl5ibrcom 237 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (¬ 𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) ≠ 𝐶))
2120necon4bd 2916 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (if(𝑥𝑆, 𝐶, 𝐵) = 𝐶𝑥𝑆))
2215, 21impbid2 216 . . . . . . . 8 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝑥𝑆 ↔ if(𝑥𝑆, 𝐶, 𝐵) = 𝐶))
23 simplrr 820 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
2423fveq1d 6306 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝐺𝑥) = ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥))
25 id 22 . . . . . . . . . . 11 (𝑥𝐴𝑥𝐴)
263, 6ifcld 4239 . . . . . . . . . . . 12 (𝜑 → if(𝑥𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
2726adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → if(𝑥𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
28 eleq1 2791 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦𝑆𝑥𝑆))
2928ifbid 4216 . . . . . . . . . . . 12 (𝑦 = 𝑥 → if(𝑦𝑆, 𝐶, 𝐵) = if(𝑥𝑆, 𝐶, 𝐵))
3029, 9fvmptg 6394 . . . . . . . . . . 11 ((𝑥𝐴 ∧ if(𝑥𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶}) → ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3125, 27, 30syl2anr 496 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3224, 31eqtrd 2758 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝐺𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3332eqeq1d 2726 . . . . . . . 8 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → ((𝐺𝑥) = 𝐶 ↔ if(𝑥𝑆, 𝐶, 𝐵) = 𝐶))
3422, 33bitr4d 271 . . . . . . 7 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝑥𝑆 ↔ (𝐺𝑥) = 𝐶))
3534pm5.32da 676 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → ((𝑥𝐴𝑥𝑆) ↔ (𝑥𝐴 ∧ (𝐺𝑥) = 𝐶)))
36 simprl 811 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝑆𝐴)
3736sseld 3708 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆𝑥𝐴))
3837pm4.71rd 670 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆 ↔ (𝑥𝐴𝑥𝑆)))
39 ffn 6158 . . . . . . . 8 (𝐺:𝐴⟶{𝐵, 𝐶} → 𝐺 Fn 𝐴)
4014, 39syl 17 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺 Fn 𝐴)
41 fniniseg 6453 . . . . . . 7 (𝐺 Fn 𝐴 → (𝑥 ∈ (𝐺 “ {𝐶}) ↔ (𝑥𝐴 ∧ (𝐺𝑥) = 𝐶)))
4240, 41syl 17 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ (𝐺 “ {𝐶}) ↔ (𝑥𝐴 ∧ (𝐺𝑥) = 𝐶)))
4335, 38, 423bitr4d 300 . . . . 5 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆𝑥 ∈ (𝐺 “ {𝐶})))
4443eqrdv 2722 . . . 4 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝑆 = (𝐺 “ {𝐶}))
4514, 44jca 555 . . 3 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶})))
46 simprr 813 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝑆 = (𝐺 “ {𝐶}))
47 cnvimass 5595 . . . . . 6 (𝐺 “ {𝐶}) ⊆ dom 𝐺
48 fdm 6164 . . . . . . 7 (𝐺:𝐴⟶{𝐵, 𝐶} → dom 𝐺 = 𝐴)
4948ad2antrl 766 . . . . . 6 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → dom 𝐺 = 𝐴)
5047, 49syl5sseq 3759 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝐺 “ {𝐶}) ⊆ 𝐴)
5146, 50eqsstrd 3745 . . . 4 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝑆𝐴)
5239ad2antrl 766 . . . . . 6 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺 Fn 𝐴)
53 dffn5 6355 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑦𝐴 ↦ (𝐺𝑦)))
5452, 53sylib 208 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺 = (𝑦𝐴 ↦ (𝐺𝑦)))
55 simplrr 820 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → 𝑆 = (𝐺 “ {𝐶}))
5655eleq2d 2789 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦𝑆𝑦 ∈ (𝐺 “ {𝐶})))
57 fniniseg 6453 . . . . . . . . . . . 12 (𝐺 Fn 𝐴 → (𝑦 ∈ (𝐺 “ {𝐶}) ↔ (𝑦𝐴 ∧ (𝐺𝑦) = 𝐶)))
5852, 57syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝑦 ∈ (𝐺 “ {𝐶}) ↔ (𝑦𝐴 ∧ (𝐺𝑦) = 𝐶)))
5958baibd 986 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦 ∈ (𝐺 “ {𝐶}) ↔ (𝐺𝑦) = 𝐶))
6056, 59bitrd 268 . . . . . . . . 9 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦𝑆 ↔ (𝐺𝑦) = 𝐶))
6160biimpa 502 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ 𝑦𝑆) → (𝐺𝑦) = 𝐶)
62 iftrue 4200 . . . . . . . . 9 (𝑦𝑆 → if(𝑦𝑆, 𝐶, 𝐵) = 𝐶)
6362adantl 473 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ 𝑦𝑆) → if(𝑦𝑆, 𝐶, 𝐵) = 𝐶)
6461, 63eqtr4d 2761 . . . . . . 7 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ 𝑦𝑆) → (𝐺𝑦) = if(𝑦𝑆, 𝐶, 𝐵))
65 simprl 811 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺:𝐴⟶{𝐵, 𝐶})
6665ffvelrnda 6474 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝐺𝑦) ∈ {𝐵, 𝐶})
67 fvex 6314 . . . . . . . . . . . . . 14 (𝐺𝑦) ∈ V
6867elpr 4306 . . . . . . . . . . . . 13 ((𝐺𝑦) ∈ {𝐵, 𝐶} ↔ ((𝐺𝑦) = 𝐵 ∨ (𝐺𝑦) = 𝐶))
6966, 68sylib 208 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → ((𝐺𝑦) = 𝐵 ∨ (𝐺𝑦) = 𝐶))
7069ord 391 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (¬ (𝐺𝑦) = 𝐵 → (𝐺𝑦) = 𝐶))
7170, 60sylibrd 249 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (¬ (𝐺𝑦) = 𝐵𝑦𝑆))
7271con1d 139 . . . . . . . . 9 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (¬ 𝑦𝑆 → (𝐺𝑦) = 𝐵))
7372imp 444 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝑆) → (𝐺𝑦) = 𝐵)
74 iffalse 4203 . . . . . . . . 9 𝑦𝑆 → if(𝑦𝑆, 𝐶, 𝐵) = 𝐵)
7574adantl 473 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝑆) → if(𝑦𝑆, 𝐶, 𝐵) = 𝐵)
7673, 75eqtr4d 2761 . . . . . . 7 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝑆) → (𝐺𝑦) = if(𝑦𝑆, 𝐶, 𝐵))
7764, 76pm2.61dan 867 . . . . . 6 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝐺𝑦) = if(𝑦𝑆, 𝐶, 𝐵))
7877mpteq2dva 4852 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝑦𝐴 ↦ (𝐺𝑦)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
7954, 78eqtrd 2758 . . . 4 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
8051, 79jca 555 . . 3 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))))
8145, 80impbida 913 . 2 (𝜑 → ((𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))))
82 pw2f1o.1 . . . 4 (𝜑𝐴𝑉)
83 elpw2g 4932 . . . 4 (𝐴𝑉 → (𝑆 ∈ 𝒫 𝐴𝑆𝐴))
8482, 83syl 17 . . 3 (𝜑 → (𝑆 ∈ 𝒫 𝐴𝑆𝐴))
85 eleq1 2791 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑆𝑦𝑆))
8685ifbid 4216 . . . . . 6 (𝑧 = 𝑦 → if(𝑧𝑆, 𝐶, 𝐵) = if(𝑦𝑆, 𝐶, 𝐵))
8786cbvmptv 4858 . . . . 5 (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))
8887a1i 11 . . . 4 (𝜑 → (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
8988eqeq2d 2734 . . 3 (𝜑 → (𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵)) ↔ 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))))
9084, 89anbi12d 749 . 2 (𝜑 → ((𝑆 ∈ 𝒫 𝐴𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵))) ↔ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))))
91 prex 5014 . . . 4 {𝐵, 𝐶} ∈ V
92 elmapg 7987 . . . 4 (({𝐵, 𝐶} ∈ V ∧ 𝐴𝑉) → (𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
9391, 82, 92sylancr 698 . . 3 (𝜑 → (𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
9493anbi1d 743 . 2 (𝜑 → ((𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑆 = (𝐺 “ {𝐶})) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))))
9581, 90, 943bitr4d 300 1 (𝜑 → ((𝑆 ∈ 𝒫 𝐴𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵))) ↔ (𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑆 = (𝐺 “ {𝐶}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1596  wcel 2103  wne 2896  Vcvv 3304  wss 3680  ifcif 4194  𝒫 cpw 4266  {csn 4285  {cpr 4287  cmpt 4837  ccnv 5217  dom cdm 5218  cima 5221   Fn wfn 5996  wf 5997  cfv 6001  (class class class)co 6765  𝑚 cmap 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-map 7976
This theorem is referenced by:  pw2f1o  8181  sqff1o  25028
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