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Theorem pw2f1olem 8009

Description: Lemma for pw2f1o 8010. (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 4271	. . . . . . . . . 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 4270	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵, 𝐶})
6	4, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶})
7	3, 6	ifcld 4108	. . . . . . . 8 ⊢ (𝜑 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
8	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
9		eqid 2626	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
10	8, 9	fmptd 6341	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
11	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
12		simprr 795	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
13	12	feq1d 5989	. . . . 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}))
14	11, 13	mpbird 247	. . . 4 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺:𝐴⟶{𝐵, 𝐶})
15		iftrue 4069	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
16		pw2f1o.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
17	16	ad2antrr 761	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 𝐶)
18		iffalse 4072	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
19	18	neeq1d 2855	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝑆 → (if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
20	17, 19	syl5ibrcom 237	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ≠ 𝐶))
21	20	necon4bd 2816	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶 → 𝑥 ∈ 𝑆))
22	15, 21	impbid2 216	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↔ if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶))
23		simplrr 800	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
24	23	fveq1d 6152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = ((𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))‘𝑥))
25		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
26	3, 6	ifcld 4108	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
27	26	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
28		eleq1 2692	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
29	28	ifbid 4085	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
30	29, 9	fvmptg 6238	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶}) → ((𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))‘𝑥) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
31	25, 27, 30	syl2anr 495	. . . . . . . . . 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	24, 31	eqtrd 2660	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
33	32	eqeq1d 2628	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = 𝐶 ↔ if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶))
34	22, 33	bitr4d 271	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↔ (𝐺‘𝑥) = 𝐶))
35	34	pm5.32da 672	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) = 𝐶)))
36		simprl 793	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝑆 ⊆ 𝐴)
37	36	sseld 3587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴))
38	37	pm4.71rd 666	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆)))
39		ffn 6004	. . . . . . . 8 ⊢ (𝐺:𝐴⟶{𝐵, 𝐶} → 𝐺 Fn 𝐴)
40	14, 39	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺 Fn 𝐴)
41		fniniseg 6295	. . . . . . 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 300	. . . . 5 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (^◡𝐺 “ {𝐶})))
44	43	eqrdv 2624	. . . 4 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝑆 = (^◡𝐺 “ {𝐶}))
45	14, 44	jca 554	. . 3 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶})))
46		simprr 795	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → 𝑆 = (^◡𝐺 “ {𝐶}))
47		cnvimass 5448	. . . . . 6 ⊢ (^◡𝐺 “ {𝐶}) ⊆ dom 𝐺
48		fdm 6010	. . . . . . 7 ⊢ (𝐺:𝐴⟶{𝐵, 𝐶} → dom 𝐺 = 𝐴)
49	48	ad2antrl 763	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → dom 𝐺 = 𝐴)
50	47, 49	syl5sseq 3637	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → (^◡𝐺 “ {𝐶}) ⊆ 𝐴)
51	46, 50	eqsstrd 3623	. . . 4 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → 𝑆 ⊆ 𝐴)
52	39	ad2antrl 763	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → 𝐺 Fn 𝐴)
53		dffn5 6199	. . . . . 6 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)))
54	52, 53	sylib 208	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)))
55		simplrr 800	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → 𝑆 = (^◡𝐺 “ {𝐶}))
56	55	eleq2d 2689	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑆 ↔ 𝑦 ∈ (^◡𝐺 “ {𝐶})))
57		fniniseg 6295	. . . . . . . . . . . 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 947	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ (^◡𝐺 “ {𝐶}) ↔ (𝐺‘𝑦) = 𝐶))
60	56, 59	bitrd 268	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑆 ↔ (𝐺‘𝑦) = 𝐶))
61	60	biimpa 501	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = 𝐶)
62		iftrue 4069	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
63	62	adantl 482	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
64	61, 63	eqtr4d 2663	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
65		simprl 793	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → 𝐺:𝐴⟶{𝐵, 𝐶})
66	65	ffvelrnda 6316	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ {𝐵, 𝐶})
67		fvex 6160	. . . . . . . . . . . . . 14 ⊢ (𝐺‘𝑦) ∈ V
68	67	elpr 4174	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑦) ∈ {𝐵, 𝐶} ↔ ((𝐺‘𝑦) = 𝐵 ∨ (𝐺‘𝑦) = 𝐶))
69	66, 68	sylib 208	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑦) = 𝐵 ∨ (𝐺‘𝑦) = 𝐶))
70	69	ord 392	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ (𝐺‘𝑦) = 𝐵 → (𝐺‘𝑦) = 𝐶))
71	70, 60	sylibrd 249	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ (𝐺‘𝑦) = 𝐵 → 𝑦 ∈ 𝑆))
72	71	con1d 139	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ∈ 𝑆 → (𝐺‘𝑦) = 𝐵))
73	72	imp 445	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = 𝐵)
74		iffalse 4072	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ 𝑆 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
75	74	adantl 482	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
76	73, 75	eqtr4d 2663	. . . . . . 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 831	. . . . . 6 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
78	77	mpteq2dva 4709	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
79	54, 78	eqtrd 2660	. . . 4 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
80	51, 79	jca 554	. . 3 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))) → (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))))
81	45, 80	impbida 876	. 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 4792	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
84	82, 83	syl 17	. . 3 ⊢ (𝜑 → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
85		eleq1 2692	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
86	85	ifbid 4085	. . . . . 6 ⊢ (𝑧 = 𝑦 → if(𝑧 ∈ 𝑆, 𝐶, 𝐵) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
87	86	cbvmptv 4715	. . . . 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 2636	. . 3 ⊢ (𝜑 → (𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵)) ↔ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))))
90	84, 89	anbi12d 746	. 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 4875	. . . 4 ⊢ {𝐵, 𝐶} ∈ V
92		elmapg 7816	. . . 4 ⊢ (({𝐵, 𝐶} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐺 ∈ ({𝐵, 𝐶} ↑_𝑚 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
93	91, 82, 92	sylancr 694	. . 3 ⊢ (𝜑 → (𝐺 ∈ ({𝐵, 𝐶} ↑_𝑚 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
94	93	anbi1d 740	. 2 ⊢ (𝜑 → ((𝐺 ∈ ({𝐵, 𝐶} ↑_𝑚 𝐴) ∧ 𝑆 = (^◡𝐺 “ {𝐶})) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (^◡𝐺 “ {𝐶}))))
95	81, 90, 94	3bitr4d 300	1 ⊢ (𝜑 → ((𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵))) ↔ (𝐺 ∈ ({𝐵, 𝐶} ↑_𝑚 𝐴) ∧ 𝑆 = (^◡𝐺 “ {𝐶}))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1480 ∈ wcel 1992 ≠ wne 2796 Vcvv 3191 ⊆ wss 3560 ifcif 4063 𝒫 cpw 4135 {csn 4153 {cpr 4155 ↦ cmpt 4678 ^◡ccnv 5078 dom cdm 5079 “ cima 5082 Fn wfn 5845 ⟶wf 5846 ‘cfv 5850 (class class class)co 6605 ↑_𝑚 cmap 7803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1841 ax-6 1890 ax-7 1937 ax-8 1994 ax-9 2001 ax-10 2021 ax-11 2036 ax-12 2049 ax-13 2250 ax-ext 2606 ax-sep 4746 ax-nul 4754 ax-pow 4808 ax-pr 4872 ax-un 6903

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1883 df-eu 2478 df-mo 2479 df-clab 2613 df-cleq 2619 df-clel 2622 df-nfc 2756 df-ne 2797 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3193 df-sbc 3423 df-dif 3563 df-un 3565 df-in 3567 df-ss 3574 df-nul 3897 df-if 4064 df-pw 4137 df-sn 4154 df-pr 4156 df-op 4160 df-uni 4408 df-br 4619 df-opab 4679 df-mpt 4680 df-id 4994 df-xp 5085 df-rel 5086 df-cnv 5087 df-co 5088 df-dm 5089 df-rn 5090 df-res 5091 df-ima 5092 df-iota 5813 df-fun 5852 df-fn 5853 df-f 5854 df-fv 5858 df-ov 6608 df-oprab 6609 df-mpt2 6610 df-map 7805

This theorem is referenced by: pw2f1o 8010 sqff1o 24803

W3C validator