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Theorem pw2f1olem 8306
Description: Lemma for pw2f1o 8307. (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 4485 . . . . . . . . . 10 (𝐶𝑊𝐶 ∈ {𝐵, 𝐶})
31, 2syl 17 . . . . . . . . 9 (𝜑𝐶 ∈ {𝐵, 𝐶})
4 pw2f1o.2 . . . . . . . . . 10 (𝜑𝐵𝑊)
5 prid1g 4484 . . . . . . . . . 10 (𝐵𝑊𝐵 ∈ {𝐵, 𝐶})
64, 5syl 17 . . . . . . . . 9 (𝜑𝐵 ∈ {𝐵, 𝐶})
73, 6ifcld 4322 . . . . . . . 8 (𝜑 → if(𝑦𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
87adantr 473 . . . . . . 7 ((𝜑𝑦𝐴) → if(𝑦𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
98fmpttd 6611 . . . . . 6 (𝜑 → (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
109adantr 473 . . . . 5 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
11 simprr 790 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
1211feq1d 6241 . . . . 5 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ↔ (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶}))
1310, 12mpbird 249 . . . 4 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺:𝐴⟶{𝐵, 𝐶})
14 iftrue 4283 . . . . . . . . 9 (𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) = 𝐶)
15 pw2f1o.4 . . . . . . . . . . . 12 (𝜑𝐵𝐶)
1615ad2antrr 718 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → 𝐵𝐶)
17 iffalse 4286 . . . . . . . . . . . 12 𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) = 𝐵)
1817neeq1d 3030 . . . . . . . . . . 11 𝑥𝑆 → (if(𝑥𝑆, 𝐶, 𝐵) ≠ 𝐶𝐵𝐶))
1916, 18syl5ibrcom 239 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (¬ 𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) ≠ 𝐶))
2019necon4bd 2991 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (if(𝑥𝑆, 𝐶, 𝐵) = 𝐶𝑥𝑆))
2114, 20impbid2 218 . . . . . . . 8 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝑥𝑆 ↔ if(𝑥𝑆, 𝐶, 𝐵) = 𝐶))
22 simplrr 797 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
2322fveq1d 6413 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝐺𝑥) = ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥))
24 id 22 . . . . . . . . . . 11 (𝑥𝐴𝑥𝐴)
253, 6ifcld 4322 . . . . . . . . . . . 12 (𝜑 → if(𝑥𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
2625adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → if(𝑥𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
27 eleq1w 2861 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦𝑆𝑥𝑆))
2827ifbid 4299 . . . . . . . . . . . 12 (𝑦 = 𝑥 → if(𝑦𝑆, 𝐶, 𝐵) = if(𝑥𝑆, 𝐶, 𝐵))
29 eqid 2799 . . . . . . . . . . . 12 (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))
3028, 29fvmptg 6505 . . . . . . . . . . 11 ((𝑥𝐴 ∧ if(𝑥𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶}) → ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3124, 26, 30syl2anr 591 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3223, 31eqtrd 2833 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝐺𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3332eqeq1d 2801 . . . . . . . 8 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → ((𝐺𝑥) = 𝐶 ↔ if(𝑥𝑆, 𝐶, 𝐵) = 𝐶))
3421, 33bitr4d 274 . . . . . . 7 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝑥𝑆 ↔ (𝐺𝑥) = 𝐶))
3534pm5.32da 575 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → ((𝑥𝐴𝑥𝑆) ↔ (𝑥𝐴 ∧ (𝐺𝑥) = 𝐶)))
36 simprl 788 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝑆𝐴)
3736sseld 3797 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆𝑥𝐴))
3837pm4.71rd 559 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆 ↔ (𝑥𝐴𝑥𝑆)))
39 ffn 6256 . . . . . . . 8 (𝐺:𝐴⟶{𝐵, 𝐶} → 𝐺 Fn 𝐴)
4013, 39syl 17 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺 Fn 𝐴)
41 fniniseg 6564 . . . . . . 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 303 . . . . 5 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆𝑥 ∈ (𝐺 “ {𝐶})))
4443eqrdv 2797 . . . 4 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝑆 = (𝐺 “ {𝐶}))
4513, 44jca 508 . . 3 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶})))
46 simprr 790 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝑆 = (𝐺 “ {𝐶}))
47 cnvimass 5702 . . . . . 6 (𝐺 “ {𝐶}) ⊆ dom 𝐺
48 fdm 6264 . . . . . . 7 (𝐺:𝐴⟶{𝐵, 𝐶} → dom 𝐺 = 𝐴)
4948ad2antrl 720 . . . . . 6 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → dom 𝐺 = 𝐴)
5047, 49syl5sseq 3849 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝐺 “ {𝐶}) ⊆ 𝐴)
5146, 50eqsstrd 3835 . . . 4 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝑆𝐴)
5239ad2antrl 720 . . . . . 6 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺 Fn 𝐴)
53 dffn5 6466 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑦𝐴 ↦ (𝐺𝑦)))
5452, 53sylib 210 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺 = (𝑦𝐴 ↦ (𝐺𝑦)))
55 simplrr 797 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → 𝑆 = (𝐺 “ {𝐶}))
5655eleq2d 2864 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦𝑆𝑦 ∈ (𝐺 “ {𝐶})))
57 fniniseg 6564 . . . . . . . . . . . 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 536 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦 ∈ (𝐺 “ {𝐶}) ↔ (𝐺𝑦) = 𝐶))
6056, 59bitrd 271 . . . . . . . . 9 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦𝑆 ↔ (𝐺𝑦) = 𝐶))
6160biimpa 469 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ 𝑦𝑆) → (𝐺𝑦) = 𝐶)
62 iftrue 4283 . . . . . . . . 9 (𝑦𝑆 → if(𝑦𝑆, 𝐶, 𝐵) = 𝐶)
6362adantl 474 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ 𝑦𝑆) → if(𝑦𝑆, 𝐶, 𝐵) = 𝐶)
6461, 63eqtr4d 2836 . . . . . . 7 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ 𝑦𝑆) → (𝐺𝑦) = if(𝑦𝑆, 𝐶, 𝐵))
65 simprl 788 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺:𝐴⟶{𝐵, 𝐶})
6665ffvelrnda 6585 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝐺𝑦) ∈ {𝐵, 𝐶})
67 fvex 6424 . . . . . . . . . . . . . 14 (𝐺𝑦) ∈ V
6867elpr 4391 . . . . . . . . . . . . 13 ((𝐺𝑦) ∈ {𝐵, 𝐶} ↔ ((𝐺𝑦) = 𝐵 ∨ (𝐺𝑦) = 𝐶))
6966, 68sylib 210 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → ((𝐺𝑦) = 𝐵 ∨ (𝐺𝑦) = 𝐶))
7069ord 891 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (¬ (𝐺𝑦) = 𝐵 → (𝐺𝑦) = 𝐶))
7170, 60sylibrd 251 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (¬ (𝐺𝑦) = 𝐵𝑦𝑆))
7271con1d 142 . . . . . . . . 9 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (¬ 𝑦𝑆 → (𝐺𝑦) = 𝐵))
7372imp 396 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝑆) → (𝐺𝑦) = 𝐵)
74 iffalse 4286 . . . . . . . . 9 𝑦𝑆 → if(𝑦𝑆, 𝐶, 𝐵) = 𝐵)
7574adantl 474 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝑆) → if(𝑦𝑆, 𝐶, 𝐵) = 𝐵)
7673, 75eqtr4d 2836 . . . . . . 7 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝑆) → (𝐺𝑦) = if(𝑦𝑆, 𝐶, 𝐵))
7764, 76pm2.61dan 848 . . . . . 6 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝐺𝑦) = if(𝑦𝑆, 𝐶, 𝐵))
7877mpteq2dva 4937 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝑦𝐴 ↦ (𝐺𝑦)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
7954, 78eqtrd 2833 . . . 4 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
8051, 79jca 508 . . 3 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))))
8145, 80impbida 836 . 2 (𝜑 → ((𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))))
82 pw2f1o.1 . . . 4 (𝜑𝐴𝑉)
83 elpw2g 5019 . . . 4 (𝐴𝑉 → (𝑆 ∈ 𝒫 𝐴𝑆𝐴))
8482, 83syl 17 . . 3 (𝜑 → (𝑆 ∈ 𝒫 𝐴𝑆𝐴))
85 eleq1w 2861 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑆𝑦𝑆))
8685ifbid 4299 . . . . . 6 (𝑧 = 𝑦 → if(𝑧𝑆, 𝐶, 𝐵) = if(𝑦𝑆, 𝐶, 𝐵))
8786cbvmptv 4943 . . . . 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 2809 . . 3 (𝜑 → (𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵)) ↔ 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))))
9084, 89anbi12d 625 . 2 (𝜑 → ((𝑆 ∈ 𝒫 𝐴𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵))) ↔ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))))
91 prex 5100 . . . 4 {𝐵, 𝐶} ∈ V
92 elmapg 8108 . . . 4 (({𝐵, 𝐶} ∈ V ∧ 𝐴𝑉) → (𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
9391, 82, 92sylancr 582 . . 3 (𝜑 → (𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
9493anbi1d 624 . 2 (𝜑 → ((𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑆 = (𝐺 “ {𝐶})) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))))
9581, 90, 943bitr4d 303 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 198  wa 385  wo 874   = wceq 1653  wcel 2157  wne 2971  Vcvv 3385  wss 3769  ifcif 4277  𝒫 cpw 4349  {csn 4368  {cpr 4370  cmpt 4922  ccnv 5311  dom cdm 5312  cima 5315   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6878  𝑚 cmap 8095
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 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
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 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097
This theorem is referenced by:  pw2f1o  8307  sqff1o  25260
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