Theorem pw2f1odclem 6890

Description: Lemma for pw2f1odc 6891. (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)
pw2f1odc.4	⊢ (𝜑 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞)

Assertion

Ref	Expression
pw2f1odclem	⊢ (𝜑 → ((𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵))) ↔ (𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑆 = (◡𝐺 “ {𝐶}))))

Distinct variable groups:   𝐴,𝑝,𝑞   𝑧,𝐴   𝑧,𝐵   𝑧,𝐶   𝑆,𝑞   𝑧,𝑆

Allowed substitution hints:   𝜑(𝑧,𝑞,𝑝)   𝐵(𝑞,𝑝)   𝐶(𝑞,𝑝)   𝑆(𝑝)   𝐺(𝑧,𝑞,𝑝)   𝑉(𝑧,𝑞,𝑝)   𝑊(𝑧,𝑞,𝑝)

Proof of Theorem pw2f1odclem

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 3723	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ {𝐵, 𝐶})
3	1, 2	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ {𝐵, 𝐶})
4	3	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐶 ∈ {𝐵, 𝐶})
5		pw2f1o.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6		prid1g 3722	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵, 𝐶})
7	5, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶})
8	7	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ {𝐵, 𝐶})
9		eleq2 2257	. . . . . . . . . 10 ⊢ (𝑞 = 𝑆 → (𝑦 ∈ 𝑞 ↔ 𝑦 ∈ 𝑆))
10	9	dcbid 839	. . . . . . . . 9 ⊢ (𝑞 = 𝑆 → (DECID 𝑦 ∈ 𝑞 ↔ DECID 𝑦 ∈ 𝑆))
11		elequ1 2168	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑦 → (𝑝 ∈ 𝑞 ↔ 𝑦 ∈ 𝑞))
12	11	dcbid 839	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑦 → (DECID 𝑝 ∈ 𝑞 ↔ DECID 𝑦 ∈ 𝑞))
13	12	ralbidv 2494	. . . . . . . . . 10 ⊢ (𝑝 = 𝑦 → (∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞 ↔ ∀𝑞 ∈ 𝒫 𝐴DECID 𝑦 ∈ 𝑞))
14		pw2f1odc.4	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞)
15	14	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞)
16		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
17	13, 15, 16	rspcdva 2869	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → ∀𝑞 ∈ 𝒫 𝐴DECID 𝑦 ∈ 𝑞)
18		simplr 528	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑆 ⊆ 𝐴)
19		pw2f1o.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
20	19	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐴 ∈ 𝑉)
21	20, 18	ssexd 4169	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑆 ∈ V)
22		elpwg 3609	. . . . . . . . . . 11 ⊢ (𝑆 ∈ V → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
23	21, 22	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
24	18, 23	mpbird 167	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑆 ∈ 𝒫 𝐴)
25	10, 17, 24	rspcdva 2869	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → DECID 𝑦 ∈ 𝑆)
26	4, 8, 25	ifcldcd 3593	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐴) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
27	26	fmpttd 5713	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ⊆ 𝐴) → (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
28	27	adantrr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)):𝐴⟶{𝐵, 𝐶})
29		simprr 531	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
30	29	feq1d 5390	. . . . 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}))
31	28, 30	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝐺:𝐴⟶{𝐵, 𝐶})
32		iftrue 3562	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
33		eleq2 2257	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑆 → (𝑥 ∈ 𝑞 ↔ 𝑥 ∈ 𝑆))
34	33	dcbid 839	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑆 → (DECID 𝑥 ∈ 𝑞 ↔ DECID 𝑥 ∈ 𝑆))
35		elequ1 2168	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑥 → (𝑝 ∈ 𝑞 ↔ 𝑥 ∈ 𝑞))
36	35	dcbid 839	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑥 → (DECID 𝑝 ∈ 𝑞 ↔ DECID 𝑥 ∈ 𝑞))
37	36	ralbidv 2494	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑥 → (∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞 ↔ ∀𝑞 ∈ 𝒫 𝐴DECID 𝑥 ∈ 𝑞))
38	14	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝒫 𝐴DECID 𝑝 ∈ 𝑞)
39		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
40	37, 38, 39	rspcdva 2869	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → ∀𝑞 ∈ 𝒫 𝐴DECID 𝑥 ∈ 𝑞)
41		simplrl 535	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝑆 ⊆ 𝐴)
42	19	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ 𝑉)
43	42, 41	ssexd 4169	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ V)
44	43, 22	syl 14	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
45	41, 44	mpbird 167	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝒫 𝐴)
46	34, 40, 45	rspcdva 2869	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → DECID 𝑥 ∈ 𝑆)
47		pw2f1o.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
48	47	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 𝐶)
49		iffalse 3565	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
50	49	neeq1d 2382	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 𝑆 → (if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))
51	48, 50	syl5ibrcom 157	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ≠ 𝐶))
52	51	a1d 22	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (DECID 𝑥 ∈ 𝑆 → (¬ 𝑥 ∈ 𝑆 → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ≠ 𝐶)))
53	52	necon4bddc 2435	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (DECID 𝑥 ∈ 𝑆 → (if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶 → 𝑥 ∈ 𝑆)))
54	46, 53	mpd 13	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶 → 𝑥 ∈ 𝑆))
55	32, 54	impbid2 143	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↔ if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶))
56		simplrr 536	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
57	56	fveq1d 5556	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = ((𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))‘𝑥))
58		eqid 2193	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
59		eleq1w 2254	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆))
60	59	ifbid 3578	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
61	3	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ {𝐵, 𝐶})
62	7	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝐵, 𝐶})
63	61, 62, 46	ifcldcd 3593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝑆, 𝐶, 𝐵) ∈ {𝐵, 𝐶})
64	58, 60, 39, 63	fvmptd3 5651	. . . . . . . . . 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))
65	57, 64	eqtrd 2226	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = if(𝑥 ∈ 𝑆, 𝐶, 𝐵))
66	65	eqeq1d 2202	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = 𝐶 ↔ if(𝑥 ∈ 𝑆, 𝐶, 𝐵) = 𝐶))
67	55, 66	bitr4d 191	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↔ (𝐺‘𝑥) = 𝐶))
68	67	pm5.32da 452	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) = 𝐶)))
69		simprl 529	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝑆 ⊆ 𝐴)
70	69	sseld 3178	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴))
71	70	pm4.71rd 394	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝑆)))
72		ffn 5403	. . . . . . 7 ⊢ (𝐺:𝐴⟶{𝐵, 𝐶} → 𝐺 Fn 𝐴)
73		fniniseg 5678	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → (𝑥 ∈ (◡𝐺 “ {𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) = 𝐶)))
74	31, 72, 73	3syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ (◡𝐺 “ {𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐺‘𝑥) = 𝐶)))
75	68, 71, 74	3bitr4d 220	. . . . 5 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝑥 ∈ 𝑆 ↔ 𝑥 ∈ (◡𝐺 “ {𝐶})))
76	75	eqrdv 2191	. . . 4 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → 𝑆 = (◡𝐺 “ {𝐶}))
77	31, 76	jca 306	. . 3 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))) → (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶})))
78		simprr 531	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝑆 = (◡𝐺 “ {𝐶}))
79		cnvimass 5028	. . . . . 6 ⊢ (◡𝐺 “ {𝐶}) ⊆ dom 𝐺
80		fdm 5409	. . . . . . 7 ⊢ (𝐺:𝐴⟶{𝐵, 𝐶} → dom 𝐺 = 𝐴)
81	80	ad2antrl 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → dom 𝐺 = 𝐴)
82	79, 81	sseqtrid 3229	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (◡𝐺 “ {𝐶}) ⊆ 𝐴)
83	78, 82	eqsstrd 3215	. . . 4 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝑆 ⊆ 𝐴)
84	72	ad2antrl 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺 Fn 𝐴)
85		dffn5im 5602	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)))
86	84, 85	syl 14	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)))
87		simplrr 536	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → 𝑆 = (◡𝐺 “ {𝐶}))
88	87	eleq2d 2263	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑆 ↔ 𝑦 ∈ (◡𝐺 “ {𝐶})))
89		fniniseg 5678	. . . . . . . . . . . 12 ⊢ (𝐺 Fn 𝐴 → (𝑦 ∈ (◡𝐺 “ {𝐶}) ↔ (𝑦 ∈ 𝐴 ∧ (𝐺‘𝑦) = 𝐶)))
90	84, 89	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (𝑦 ∈ (◡𝐺 “ {𝐶}) ↔ (𝑦 ∈ 𝐴 ∧ (𝐺‘𝑦) = 𝐶)))
91	90	baibd 924	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ (◡𝐺 “ {𝐶}) ↔ (𝐺‘𝑦) = 𝐶))
92	88, 91	bitrd 188	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑆 ↔ (𝐺‘𝑦) = 𝐶))
93	92	biimpa 296	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = 𝐶)
94		iftrue 3562	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
95	94	adantl 277	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐶)
96	93, 95	eqtr4d 2229	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
97	92	notbid 668	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ∈ 𝑆 ↔ ¬ (𝐺‘𝑦) = 𝐶))
98	97	biimpa 296	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → ¬ (𝐺‘𝑦) = 𝐶)
99		simprl 529	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺:𝐴⟶{𝐵, 𝐶})
100	99	ffvelcdmda 5693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) ∈ {𝐵, 𝐶})
101		elpri 3641	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑦) ∈ {𝐵, 𝐶} → ((𝐺‘𝑦) = 𝐵 ∨ (𝐺‘𝑦) = 𝐶))
102	100, 101	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑦) = 𝐵 ∨ (𝐺‘𝑦) = 𝐶))
103	102	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → ((𝐺‘𝑦) = 𝐵 ∨ (𝐺‘𝑦) = 𝐶))
104	98, 103	ecased 1360	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = 𝐵)
105		iffalse 3565	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ 𝑆 → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
106	105	adantl 277	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → if(𝑦 ∈ 𝑆, 𝐶, 𝐵) = 𝐵)
107	104, 106	eqtr4d 2229	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) ∧ ¬ 𝑦 ∈ 𝑆) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
108	83, 25	syldanl 449	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → DECID 𝑦 ∈ 𝑆)
109		exmiddc 837	. . . . . . . 8 ⊢ (DECID 𝑦 ∈ 𝑆 → (𝑦 ∈ 𝑆 ∨ ¬ 𝑦 ∈ 𝑆))
110	108, 109	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ 𝑆 ∨ ¬ 𝑦 ∈ 𝑆))
111	96, 107, 110	mpjaodan 799	. . . . . 6 ⊢ (((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
112	111	mpteq2dva 4119	. . . . 5 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (𝑦 ∈ 𝐴 ↦ (𝐺‘𝑦)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
113	86, 112	eqtrd 2226	. . . 4 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
114	83, 113	jca 306	. . 3 ⊢ ((𝜑 ∧ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))) → (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))))
115	77, 114	impbida 596	. 2 ⊢ (𝜑 → ((𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))))
116		elpw2g 4185	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
117	19, 116	syl 14	. . 3 ⊢ (𝜑 → (𝑆 ∈ 𝒫 𝐴 ↔ 𝑆 ⊆ 𝐴))
118		eleq1w 2254	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
119	118	ifbid 3578	. . . . . 6 ⊢ (𝑧 = 𝑦 → if(𝑧 ∈ 𝑆, 𝐶, 𝐵) = if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
120	119	cbvmptv 4125	. . . . 5 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))
121	120	a1i 9	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵)) = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))
122	121	eqeq2d 2205	. . 3 ⊢ (𝜑 → (𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵)) ↔ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵))))
123	117, 122	anbi12d 473	. 2 ⊢ (𝜑 → ((𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵))) ↔ (𝑆 ⊆ 𝐴 ∧ 𝐺 = (𝑦 ∈ 𝐴 ↦ if(𝑦 ∈ 𝑆, 𝐶, 𝐵)))))
124		prexg 4240	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑊) → {𝐵, 𝐶} ∈ V)
125	5, 1, 124	syl2anc 411	. . . 4 ⊢ (𝜑 → {𝐵, 𝐶} ∈ V)
126	125, 19	elmapd 6716	. . 3 ⊢ (𝜑 → (𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ↔ 𝐺:𝐴⟶{𝐵, 𝐶}))
127	126	anbi1d 465	. 2 ⊢ (𝜑 → ((𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑆 = (◡𝐺 “ {𝐶})) ↔ (𝐺:𝐴⟶{𝐵, 𝐶} ∧ 𝑆 = (◡𝐺 “ {𝐶}))))
128	115, 123, 127	3bitr4d 220	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   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  ∀wral 2472  Vcvv 2760   ⊆ wss 3153  ifcif 3557  𝒫 cpw 3601  {csn 3618  {cpr 3619   ↦ cmpt 4090  ^◡ccnv 4658  dom cdm 4659   “ cima 4662   Fn wfn 5249  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918   ↑_𝑚 cmap 6702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-map 6704

This theorem is referenced by:  pw2f1odc  6891