Theorem fliftf 5843

Description: The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
flift.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)

Assertion

Ref	Expression
fliftf	⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆))

Distinct variable groups:   𝑥,𝑅   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftf

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ Fun 𝐹) → Fun 𝐹)
2		flift.1	. . . . . . . . . . 11 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
3		flift.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
4		flift.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
5	2, 3, 4	fliftel 5837	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
6	5	exbidv 1836	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
7	6	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
8		rexcom4 2783	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵))
9		19.42v 1918	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵))
10		elisset 2774	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑆 → ∃𝑧 𝑧 = 𝐵)
11	4, 10	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑧 𝑧 = 𝐵)
12	11	biantrud 304	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 = 𝐴 ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵)))
13	9, 12	bitr4id 199	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ 𝑦 = 𝐴))
14	13	rexbidva 2491	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
15	14	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
16	8, 15	bitr3id 194	. . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
17	7, 16	bitrd 188	. . . . . . 7 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
18	17	abbidv 2311	. . . . . 6 ⊢ ((𝜑 ∧ Fun 𝐹) → {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧} = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴})
19		df-dm 4670	. . . . . 6 ⊢ dom 𝐹 = {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧}
20		eqid 2193	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
21	20	rnmpt 4911	. . . . . 6 ⊢ ran (𝑥 ∈ 𝑋 ↦ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴}
22	18, 19, 21	3eqtr4g 2251	. . . . 5 ⊢ ((𝜑 ∧ Fun 𝐹) → dom 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 𝐴))
23		df-fn 5258	. . . . 5 ⊢ (𝐹 Fn ran (𝑥 ∈ 𝑋 ↦ 𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 𝐴)))
24	1, 22, 23	sylanbrc 417	. . . 4 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹 Fn ran (𝑥 ∈ 𝑋 ↦ 𝐴))
25	2, 3, 4	fliftrel 5836	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆))
26	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27		rnss 4893	. . . . . 6 ⊢ (𝐹 ⊆ (𝑅 × 𝑆) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
28	26, 27	syl 14	. . . . 5 ⊢ ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
29		rnxpss 5098	. . . . 5 ⊢ ran (𝑅 × 𝑆) ⊆ 𝑆
30	28, 29	sstrdi 3192	. . . 4 ⊢ ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ 𝑆)
31		df-f 5259	. . . 4 ⊢ (𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆 ↔ (𝐹 Fn ran (𝑥 ∈ 𝑋 ↦ 𝐴) ∧ ran 𝐹 ⊆ 𝑆))
32	24, 30, 31	sylanbrc 417	. . 3 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆)
33	32	ex 115	. 2 ⊢ (𝜑 → (Fun 𝐹 → 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆))
34		ffun 5407	. 2 ⊢ (𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆 → Fun 𝐹)
35	33, 34	impbid1 142	1 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  {cab 2179  ∃wrex 2473   ⊆ wss 3154  ⟨cop 3622   class class class wbr 4030   ↦ cmpt 4091   × cxp 4658  dom cdm 4660  ran crn 4661  Fun wfun 5249   Fn wfn 5250  ⟶wf 5251

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-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-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  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-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263

This theorem is referenced by:  qliftf  6676