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Theorem pw2f1olem 9078
Description: Lemma for pw2f1o 9079. (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 4765 . . . . . . . . . 10 (𝐶𝑊𝐶 ∈ {𝐵, 𝐶})
31, 2syl 17 . . . . . . . . 9 (𝜑𝐶 ∈ {𝐵, 𝐶})
4 pw2f1o.2 . . . . . . . . . 10 (𝜑𝐵𝑊)
5 prid1g 4764 . . . . . . . . . 10 (𝐵𝑊𝐵 ∈ {𝐵, 𝐶})
64, 5syl 17 . . . . . . . . 9 (𝜑𝐵 ∈ {𝐵, 𝐶})
73, 6ifcld 4574 . . . . . . . 8 (𝜑 → if(𝑦𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
87adantr 481 . . . . . . 7 ((𝜑𝑦𝐴) → if(𝑦𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
98fmpttd 7116 . . . . . 6 (𝜑 → (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
109adantr 481 . . . . 5 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
11 simprr 771 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
1211feq1d 6702 . . . . 5 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ↔ (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶}))
1310, 12mpbird 256 . . . 4 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺:𝐴⟶{𝐵, 𝐶})
14 iftrue 4534 . . . . . . . . 9 (𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) = 𝐶)
15 pw2f1o.4 . . . . . . . . . . . 12 (𝜑𝐵𝐶)
1615ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → 𝐵𝐶)
17 iffalse 4537 . . . . . . . . . . . 12 𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) = 𝐵)
1817neeq1d 3000 . . . . . . . . . . 11 𝑥𝑆 → (if(𝑥𝑆, 𝐶, 𝐵) ≠ 𝐶𝐵𝐶))
1916, 18syl5ibrcom 246 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (¬ 𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) ≠ 𝐶))
2019necon4bd 2960 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (if(𝑥𝑆, 𝐶, 𝐵) = 𝐶𝑥𝑆))
2114, 20impbid2 225 . . . . . . . 8 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝑥𝑆 ↔ if(𝑥𝑆, 𝐶, 𝐵) = 𝐶))
22 simplrr 776 . . . . . . . . . . 11 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
2322fveq1d 6893 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝐺𝑥) = ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥))
24 id 22 . . . . . . . . . . 11 (𝑥𝐴𝑥𝐴)
253, 6ifcld 4574 . . . . . . . . . . . 12 (𝜑 → if(𝑥𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
2625adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → if(𝑥𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
27 eleq1w 2816 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦𝑆𝑥𝑆))
2827ifbid 4551 . . . . . . . . . . . 12 (𝑦 = 𝑥 → if(𝑦𝑆, 𝐶, 𝐵) = if(𝑥𝑆, 𝐶, 𝐵))
29 eqid 2732 . . . . . . . . . . . 12 (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))
3028, 29fvmptg 6996 . . . . . . . . . . 11 ((𝑥𝐴 ∧ if(𝑥𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶}) → ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3124, 26, 30syl2anr 597 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3223, 31eqtrd 2772 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝐺𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3332eqeq1d 2734 . . . . . . . 8 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → ((𝐺𝑥) = 𝐶 ↔ if(𝑥𝑆, 𝐶, 𝐵) = 𝐶))
3421, 33bitr4d 281 . . . . . . 7 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝑥𝑆 ↔ (𝐺𝑥) = 𝐶))
3534pm5.32da 579 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → ((𝑥𝐴𝑥𝑆) ↔ (𝑥𝐴 ∧ (𝐺𝑥) = 𝐶)))
36 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝑆𝐴)
3736sseld 3981 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆𝑥𝐴))
3837pm4.71rd 563 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆 ↔ (𝑥𝐴𝑥𝑆)))
39 ffn 6717 . . . . . . . 8 (𝐺:𝐴⟶{𝐵, 𝐶} → 𝐺 Fn 𝐴)
4013, 39syl 17 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺 Fn 𝐴)
41 fniniseg 7061 . . . . . . 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 310 . . . . 5 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆𝑥 ∈ (𝐺 “ {𝐶})))
4443eqrdv 2730 . . . 4 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝑆 = (𝐺 “ {𝐶}))
4513, 44jca 512 . . 3 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶})))
46 simprr 771 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝑆 = (𝐺 “ {𝐶}))
47 cnvimass 6080 . . . . . 6 (𝐺 “ {𝐶}) ⊆ dom 𝐺
48 fdm 6726 . . . . . . 7 (𝐺:𝐴⟶{𝐵, 𝐶} → dom 𝐺 = 𝐴)
4948ad2antrl 726 . . . . . 6 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → dom 𝐺 = 𝐴)
5047, 49sseqtrid 4034 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝐺 “ {𝐶}) ⊆ 𝐴)
5146, 50eqsstrd 4020 . . . 4 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝑆𝐴)
5239ad2antrl 726 . . . . . 6 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺 Fn 𝐴)
53 dffn5 6950 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑦𝐴 ↦ (𝐺𝑦)))
5452, 53sylib 217 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺 = (𝑦𝐴 ↦ (𝐺𝑦)))
55 simplrr 776 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → 𝑆 = (𝐺 “ {𝐶}))
5655eleq2d 2819 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦𝑆𝑦 ∈ (𝐺 “ {𝐶})))
57 fniniseg 7061 . . . . . . . . . . . 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 540 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦 ∈ (𝐺 “ {𝐶}) ↔ (𝐺𝑦) = 𝐶))
6056, 59bitrd 278 . . . . . . . . 9 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦𝑆 ↔ (𝐺𝑦) = 𝐶))
6160biimpa 477 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ 𝑦𝑆) → (𝐺𝑦) = 𝐶)
62 iftrue 4534 . . . . . . . . 9 (𝑦𝑆 → if(𝑦𝑆, 𝐶, 𝐵) = 𝐶)
6362adantl 482 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ 𝑦𝑆) → if(𝑦𝑆, 𝐶, 𝐵) = 𝐶)
6461, 63eqtr4d 2775 . . . . . . 7 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ 𝑦𝑆) → (𝐺𝑦) = if(𝑦𝑆, 𝐶, 𝐵))
65 simprl 769 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺:𝐴⟶{𝐵, 𝐶})
6665ffvelcdmda 7086 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝐺𝑦) ∈ {𝐵, 𝐶})
67 fvex 6904 . . . . . . . . . . . . . 14 (𝐺𝑦) ∈ V
6867elpr 4651 . . . . . . . . . . . . 13 ((𝐺𝑦) ∈ {𝐵, 𝐶} ↔ ((𝐺𝑦) = 𝐵 ∨ (𝐺𝑦) = 𝐶))
6966, 68sylib 217 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → ((𝐺𝑦) = 𝐵 ∨ (𝐺𝑦) = 𝐶))
7069ord 862 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (¬ (𝐺𝑦) = 𝐵 → (𝐺𝑦) = 𝐶))
7170, 60sylibrd 258 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (¬ (𝐺𝑦) = 𝐵𝑦𝑆))
7271con1d 145 . . . . . . . . 9 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (¬ 𝑦𝑆 → (𝐺𝑦) = 𝐵))
7372imp 407 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝑆) → (𝐺𝑦) = 𝐵)
74 iffalse 4537 . . . . . . . . 9 𝑦𝑆 → if(𝑦𝑆, 𝐶, 𝐵) = 𝐵)
7574adantl 482 . . . . . . . 8 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝑆) → if(𝑦𝑆, 𝐶, 𝐵) = 𝐵)
7673, 75eqtr4d 2775 . . . . . . 7 ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) ∧ ¬ 𝑦𝑆) → (𝐺𝑦) = if(𝑦𝑆, 𝐶, 𝐵))
7764, 76pm2.61dan 811 . . . . . 6 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝐺𝑦) = if(𝑦𝑆, 𝐶, 𝐵))
7877mpteq2dva 5248 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝑦𝐴 ↦ (𝐺𝑦)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
7954, 78eqtrd 2772 . . . 4 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
8051, 79jca 512 . . 3 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))))
8145, 80impbida 799 . 2 (𝜑 → ((𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))))
82 pw2f1o.1 . . . 4 (𝜑𝐴𝑉)
83 elpw2g 5344 . . . 4 (𝐴𝑉 → (𝑆 ∈ 𝒫 𝐴𝑆𝐴))
8482, 83syl 17 . . 3 (𝜑 → (𝑆 ∈ 𝒫 𝐴𝑆𝐴))
85 eleq1w 2816 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑆𝑦𝑆))
8685ifbid 4551 . . . . . 6 (𝑧 = 𝑦 → if(𝑧𝑆, 𝐶, 𝐵) = if(𝑦𝑆, 𝐶, 𝐵))
8786cbvmptv 5261 . . . . 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 2743 . . 3 (𝜑 → (𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵)) ↔ 𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))))
9084, 89anbi12d 631 . 2 (𝜑 → ((𝑆 ∈ 𝒫 𝐴𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵))) ↔ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))))
91 prex 5432 . . . 4 {𝐵, 𝐶} ∈ V
92 elmapg 8835 . . . 4 (({𝐵, 𝐶} ∈ V ∧ 𝐴𝑉) → (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
9391, 82, 92sylancr 587 . . 3 (𝜑 → (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
9493anbi1d 630 . 2 (𝜑 → ((𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑆 = (𝐺 “ {𝐶})) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))))
9581, 90, 943bitr4d 310 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 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2940  Vcvv 3474  wss 3948  ifcif 4528  𝒫 cpw 4602  {csn 4628  {cpr 4630  cmpt 5231  ccnv 5675  dom cdm 5676  cima 5679   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7411  m cmap 8822
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824
This theorem is referenced by:  pw2f1o  9079  sqff1o  26693
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