Theorem pw2f1olem 9095

Description: Lemma for pw2f1o 9096. (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 4742	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ {𝐵, 𝐶})
3	1, 2	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ {𝐵, 𝐶})
4		pw2f1o.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		prid1g 4741	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵, 𝐶})
6	4, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶})
7	3, 6	ifcld 4552	. . . . . . . 8 ⊢ (𝜑 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
8	7	adantr 480	. . . . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
11		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
12	11	feq1d 6695	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺:𝐴⟶{𝐵, 𝐶})
14		iftrue 4511	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
15		pw2f1o.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
16	15	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 𝐶)
17		iffalse 4514	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
18	17	neeq1d 2992	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝑆 → (if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
19	16, 18	syl5ibrcom 247	. . . . . . . . . 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 2953	. . . . . . . . 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 226	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↔ if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶))
22		simplrr 777	. . . . . . . . . . 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 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
25	3, 6	ifcld 4552	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
26	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
27		eleq1w 2818	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
28	27	ifbid 4529	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
29		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
30	28, 29	fvmptg 6989	. . . . . . . . . . 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 597	. . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
33	32	eqeq1d 2738	. . . . . . . 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 282	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↔ (𝐺‘𝑥) = 𝐶))
35	34	pm5.32da 579	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) = 𝐶)))
36		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝑆 ⊆ 𝐴)
37	36	sseld 3962	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴))
38	37	pm4.71rd 562	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆)))
39		ffn 6711	. . . . . . . 8 ⊢ (𝐺:𝐴⟶{𝐵, 𝐶} → 𝐺 Fn 𝐴)
40	13, 39	syl 17	. . . . . . 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 17	. . . . . 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 311	. . . . 5 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐺 “ {𝐶})))
44	43	eqrdv 2734	. . . 4 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝑆 = (◡𝐺 “ {𝐶}))
45	13, 44	jca 511	. . 3 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶})))
46		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝑆 = (◡𝐺 “ {𝐶}))
47		cnvimass 6074	. . . . . 6 ⊢ (◡𝐺 “ {𝐶}) ⊆ dom 𝐺
48		fdm 6720	. . . . . . 7 ⊢ (𝐺:𝐴⟶{𝐵, 𝐶} → dom 𝐺 = 𝐴)
49	48	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → dom 𝐺 = 𝐴)
50	47, 49	sseqtrid 4006	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (◡𝐺 “ {𝐶}) ⊆ 𝐴)
51	46, 50	eqsstrd 3998	. . . 4 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝑆 ⊆ 𝐴)
52	39	ad2antrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺 Fn 𝐴)
53		dffn5 6942	. . . . . 6 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)))
54	52, 53	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)))
55		simplrr 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → 𝑆 = (◡𝐺 “ {𝐶}))
56	55	eleq2d 2821	. . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (𝑦 ∈ (◡𝐺 “ {𝐶}) ↔ (𝑦 ∈ 𝐴 ∧ (𝐺‘𝑦) = 𝐶)))
59	58	baibd 539	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ (◡𝐺 “ {𝐶}) ↔ (𝐺‘𝑦) = 𝐶))
60	56, 59	bitrd 279	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑆 ↔ (𝐺‘𝑦) = 𝐶))
61	60	biimpa 476	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = 𝐶)
62		iftrue 4511	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
63	62	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
64	61, 63	eqtr4d 2774	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
65		simprl 770	. . . . . . . . . . . . . 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 4631	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑦) ∈ {𝐵, 𝐶} ↔ ((𝐺‘𝑦) = 𝐵 ∨ (𝐺‘𝑦) = 𝐶))
69	66, 68	sylib 218	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑦) = 𝐵 ∨ (𝐺‘𝑦) = 𝐶))
70	69	ord 864	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ (𝐺‘𝑦) = 𝐵 → (𝐺‘𝑦) = 𝐶))
71	70, 60	sylibrd 259	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ (𝐺‘𝑦) = 𝐵 → 𝑦 ∈ 𝑆))
72	71	con1d 145	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ∈ 𝑆 → (𝐺‘𝑦) = 𝐵))
73	72	imp 406	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = 𝐵)
74		iffalse 4514	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ 𝑆 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
75	74	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
76	73, 75	eqtr4d 2774	. . . . . . 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 812	. . . . . 6 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
78	77	mpteq2dva 5219	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
79	54, 78	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
80	51, 79	jca 511	. . 3 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))))
81	45, 80	impbida 800	. 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 5308	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
84	82, 83	syl 17	. . 3 ⊢ (𝜑 → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
85		eleq1w 2818	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
86	85	ifbid 4529	. . . . . 6 ⊢ (𝑧 = 𝑦 → if(𝑧 ∈ 𝑆, 𝐶, 𝐵) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
87	86	cbvmptv 5230	. . . . 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 2747	. . 3 ⊢ (𝜑 → (𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵)) ↔ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))))
90	84, 89	anbi12d 632	. 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 5412	. . . 4 ⊢ {𝐵, 𝐶} ∈ V
92		elmapg 8858	. . . 4 ⊢ (({𝐵, 𝐶} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
93	91, 82, 92	sylancr 587	. . 3 ⊢ (𝜑 → (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
94	93	anbi1d 631	. 2 ⊢ (𝜑 → ((𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑆 = (◡𝐺 “ {𝐶})) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))))
95	81, 90, 94	3bitr4d 311	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 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  Vcvv 3464   ⊆ wss 3931  ifcif 4505  𝒫 cpw 4580  {csn 4606  {cpr 4608   ↦ cmpt 5206  ^◡ccnv 5658  dom cdm 5659   “ cima 5662   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847

This theorem is referenced by:  pw2f1o  9096  sqff1o  27149