MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pw2f1olem Structured version   Visualization version   GIF version

Theorem pw2f1olem 9020
Description: Lemma for pw2f1o 9021. (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 4722 . . . . . . . . . 10 (𝐶𝑊𝐶 ∈ {𝐵, 𝐶})
31, 2syl 17 . . . . . . . . 9 (𝜑𝐶 ∈ {𝐵, 𝐶})
4 pw2f1o.2 . . . . . . . . . 10 (𝜑𝐵𝑊)
5 prid1g 4721 . . . . . . . . . 10 (𝐵𝑊𝐵 ∈ {𝐵, 𝐶})
64, 5syl 17 . . . . . . . . 9 (𝜑𝐵 ∈ {𝐵, 𝐶})
73, 6ifcld 4532 . . . . . . . 8 (𝜑 → if(𝑦𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
87adantr 481 . . . . . . 7 ((𝜑𝑦𝐴) → if(𝑦𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
98fmpttd 7063 . . . . . 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 6653 . . . . 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 4492 . . . . . . . . 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 4495 . . . . . . . . . . . 12 𝑥𝑆 → if(𝑥𝑆, 𝐶, 𝐵) = 𝐵)
1817neeq1d 3003 . . . . . . . . . . 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 2963 . . . . . . . . 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 6844 . . . . . . . . . 10 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝐺𝑥) = ((𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))‘𝑥))
24 id 22 . . . . . . . . . . 11 (𝑥𝐴𝑥𝐴)
253, 6ifcld 4532 . . . . . . . . . . . 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 2820 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦𝑆𝑥𝑆))
2827ifbid 4509 . . . . . . . . . . . 12 (𝑦 = 𝑥 → if(𝑦𝑆, 𝐶, 𝐵) = if(𝑥𝑆, 𝐶, 𝐵))
29 eqid 2736 . . . . . . . . . . . 12 (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵))
3028, 29fvmptg 6946 . . . . . . . . . . 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 2776 . . . . . . . . 9 (((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) ∧ 𝑥𝐴) → (𝐺𝑥) = if(𝑥𝑆, 𝐶, 𝐵))
3332eqeq1d 2738 . . . . . . . 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 3943 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆𝑥𝐴))
3837pm4.71rd 563 . . . . . 6 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → (𝑥𝑆 ↔ (𝑥𝐴𝑥𝑆)))
39 ffn 6668 . . . . . . . 8 (𝐺:𝐴⟶{𝐵, 𝐶} → 𝐺 Fn 𝐴)
4013, 39syl 17 . . . . . . 7 ((𝜑 ∧ (𝑆𝐴𝐺 = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))) → 𝐺 Fn 𝐴)
41 fniniseg 7010 . . . . . . 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 2734 . . . 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 6033 . . . . . 6 (𝐺 “ {𝐶}) ⊆ dom 𝐺
48 fdm 6677 . . . . . . 7 (𝐺:𝐴⟶{𝐵, 𝐶} → dom 𝐺 = 𝐴)
4948ad2antrl 726 . . . . . 6 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → dom 𝐺 = 𝐴)
5047, 49sseqtrid 3996 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝐺 “ {𝐶}) ⊆ 𝐴)
5146, 50eqsstrd 3982 . . . 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 6901 . . . . . 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 2823 . . . . . . . . . 10 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝑦𝑆𝑦 ∈ (𝐺 “ {𝐶})))
57 fniniseg 7010 . . . . . . . . . . . 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 4492 . . . . . . . . 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 2779 . . . . . . 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 7035 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) ∧ 𝑦𝐴) → (𝐺𝑦) ∈ {𝐵, 𝐶})
67 fvex 6855 . . . . . . . . . . . . . 14 (𝐺𝑦) ∈ V
6867elpr 4609 . . . . . . . . . . . . 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 4495 . . . . . . . . 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 2779 . . . . . . 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 5205 . . . . 5 ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (𝐺 “ {𝐶}))) → (𝑦𝐴 ↦ (𝐺𝑦)) = (𝑦𝐴 ↦ if(𝑦𝑆, 𝐶, 𝐵)))
7954, 78eqtrd 2776 . . . 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 5301 . . . 4 (𝐴𝑉 → (𝑆 ∈ 𝒫 𝐴𝑆𝐴))
8482, 83syl 17 . . 3 (𝜑 → (𝑆 ∈ 𝒫 𝐴𝑆𝐴))
85 eleq1w 2820 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑆𝑦𝑆))
8685ifbid 4509 . . . . . 6 (𝑧 = 𝑦 → if(𝑧𝑆, 𝐶, 𝐵) = if(𝑦𝑆, 𝐶, 𝐵))
8786cbvmptv 5218 . . . . 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 2747 . . 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 5389 . . . 4 {𝐵, 𝐶} ∈ V
92 elmapg 8778 . . . 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 2943  Vcvv 3445  wss 3910  ifcif 4486  𝒫 cpw 4560  {csn 4586  {cpr 4588  cmpt 5188  ccnv 5632  dom cdm 5633  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765
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-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767
This theorem is referenced by:  pw2f1o  9021  sqff1o  26531
  Copyright terms: Public domain W3C validator