Theorem pw2f1olem 9068

Description: Lemma for pw2f1o 9069. (Contributed by Mario Carneiro, 6-Oct-2014.)

Hypotheses

Ref	Expression
pw2f1o.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
pw2f1o.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
pw2f1o.3	⊢ (𝜑 → 𝐶 ∈ 𝑊)
pw2f1o.4	⊢ (𝜑 → 𝐵 ≠ 𝐶)

Assertion

Ref	Expression
pw2f1olem	⊢ (𝜑 → ((𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 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.

Step	Hyp	Ref	Expression
1		pw2f1o.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
2		prid2g 4726	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ {𝐵, 𝐶})
3	1, 2	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ {𝐵, 𝐶})
4		pw2f1o.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		prid1g 4725	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵, 𝐶})
6	4, 5	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶})
7	3, 6	ifcld 4533	. . . . . . . 8 ⊢ (𝜑 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
8	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
9	8	fmpttd 7110	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
10	9	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
11		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
12	11	feq1d 6687	. . . . 5 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ↔ (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶}))
13	10, 12	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺:𝐴⟶{𝐵, 𝐶})
14		iftrue 4492	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
15		pw2f1o.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
16	15	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 𝐶)
17		iffalse 4495	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
18	17	neeq1d 3015	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝑆 → (if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
19	16, 18	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ≠ 𝐶))
20	19	necon4bd 2976	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶 → 𝑥 ∈ 𝑆))
21	14, 20	impbid2 229	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↔ if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶))
22		simplrr 789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
23	22	fveq1d 6883	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = ((𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))‘𝑥))
24		id 23	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
25	3, 6	ifcld 4533	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
26	25	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
27		eleq1w 2844	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
28	27	ifbid 4510	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
29		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
30	28, 29	fvmptg 6987	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶}) → ((𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))‘𝑥) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
31	24, 26, 30	syl2anr 608	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))‘𝑥) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
32	23, 31	eqtrd 2796	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
33	32	eqeq1d 2763	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = 𝐶 ↔ if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶))
34	21, 33	bitr4d 285	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↔ (𝐺‘𝑥) = 𝐶))
35	34	pm5.32da 589	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) = 𝐶)))
36		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝑆 ⊆ 𝐴)
37	36	sseld 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴))
38	37	pm4.71rd 571	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆)))
39		ffn 6705	. . . . . . . 8 ⊢ (𝐺:𝐴⟶{𝐵, 𝐶} → 𝐺 Fn 𝐴)
40	13, 39	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺 Fn 𝐴)
41		fniniseg 7055	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → (𝑥 ∈ (◡𝐺 “ {𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) = 𝐶)))
42	40, 41	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ (◡𝐺 “ {𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) = 𝐶)))
43	35, 38, 42	3bitr4d 314	. . . . 5 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐺 “ {𝐶})))
44	43	eqrdv 2759	. . . 4 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝑆 = (◡𝐺 “ {𝐶}))
45	13, 44	jca 520	. . 3 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶})))
46		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝑆 = (◡𝐺 “ {𝐶}))
47		cnvimass 6084	. . . . . 6 ⊢ (◡𝐺 “ {𝐶}) ⊆ dom 𝐺
48		fdm 6715	. . . . . . 7 ⊢ (𝐺:𝐴⟶{𝐵, 𝐶} → dom 𝐺 = 𝐴)
49	48	ad2antrl 740	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → dom 𝐺 = 𝐴)
50	47, 49	sseqtrid 3978	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (◡𝐺 “ {𝐶}) ⊆ 𝐴)
51	46, 50	eqsstrd 3970	. . . 4 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝑆 ⊆ 𝐴)
52	39	ad2antrl 740	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺 Fn 𝐴)
53		dffn5 6939	. . . . . 6 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)))
54	52, 53	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)))
55		simplrr 789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → 𝑆 = (◡𝐺 “ {𝐶}))
56	55	eleq2d 2847	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑆 ↔ 𝑦 ∈ (◡𝐺 “ {𝐶})))
57		fniniseg 7055	. . . . . . . . . . . 12 ⊢ (𝐺 Fn 𝐴 → (𝑦 ∈ (◡𝐺 “ {𝐶}) ↔ (𝑦 ∈ 𝐴 ∧ (𝐺‘𝑦) = 𝐶)))
58	52, 57	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (𝑦 ∈ (◡𝐺 “ {𝐶}) ↔ (𝑦 ∈ 𝐴 ∧ (𝐺‘𝑦) = 𝐶)))
59	58	baibd 548	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ (◡𝐺 “ {𝐶}) ↔ (𝐺‘𝑦) = 𝐶))
60	56, 59	bitrd 282	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑆 ↔ (𝐺‘𝑦) = 𝐶))
61	60	biimpa 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = 𝐶)
62		iftrue 4492	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
63	62	adantl 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
64	61, 63	eqtr4d 2799	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
65		simprl 782	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺:𝐴⟶{𝐵, 𝐶})
66	65	ffvelcdmda 7079	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ {𝐵, 𝐶})
67		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (𝐺‘𝑦) ∈ V
68	67	elpr 4613	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑦) ∈ {𝐵, 𝐶} ↔ ((𝐺‘𝑦) = 𝐵 ∨ (𝐺‘𝑦) = 𝐶))
69	66, 68	sylib 221	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑦) = 𝐵 ∨ (𝐺‘𝑦) = 𝐶))
70	69	ord 877	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ (𝐺‘𝑦) = 𝐵 → (𝐺‘𝑦) = 𝐶))
71	70, 60	sylibrd 262	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ (𝐺‘𝑦) = 𝐵 → 𝑦 ∈ 𝑆))
72	71	con1d 146	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ∈ 𝑆 → (𝐺‘𝑦) = 𝐵))
73	72	imp 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = 𝐵)
74		iffalse 4495	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ 𝑆 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
75	74	adantl 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
76	73, 75	eqtr4d 2799	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
77	64, 76	pm2.61dan 824	. . . . . 6 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
78	77	mpteq2dva 5203	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
79	54, 78	eqtrd 2796	. . . 4 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
80	51, 79	jca 520	. . 3 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))))
81	45, 80	impbida 812	. 2 ⊢ (𝜑 → ((𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))))
82		pw2f1o.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
83		elpw2g 5303	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
84	82, 83	syl 18	. . 3 ⊢ (𝜑 → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
85		eleq1w 2844	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
86	85	ifbid 4510	. . . . . 6 ⊢ (𝑧 = 𝑦 → if(𝑧 ∈ 𝑆, 𝐶, 𝐵) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
87	86	cbvmptv 5214	. . . . 5 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
88	87	a1i 11	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
89	88	eqeq2d 2772	. . 3 ⊢ (𝜑 → (𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵)) ↔ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))))
90	84, 89	anbi12d 643	. 2 ⊢ (𝜑 → ((𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵))) ↔ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))))
91		prex 5409	. . . 4 ⊢ {𝐵, 𝐶} ∈ V
92		elmapg 8835	. . . 4 ⊢ (({𝐵, 𝐶} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
93	91, 82, 92	sylancr 598	. . 3 ⊢ (𝜑 → (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
94	93	anbi1d 642	. 2 ⊢ (𝜑 → ((𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑆 = (◡𝐺 “ {𝐶})) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))))
95	81, 90, 94	3bitr4d 314	1 ⊢ (𝜑 → ((𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵))) ↔ (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑆 = (◡𝐺 “ {𝐶}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453   ⊆ wss 3904  ifcif 4486  𝒫 cpw 4561  {csn 4588  {cpr 4590   ↦ cmpt 5191  ^◡ccnv 5660  dom cdm 5661   “ cima 5664   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8825

This theorem is referenced by:  pw2f1o  9069  sqff1o  27322