Theorem fliftf 5929

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 5923	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
6	5	exbidv 1871	. . . . . . . . 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 2823	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵))
9		19.42v 1953	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵))
10		elisset 2814	. . . . . . . . . . . . . 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 2527	. . . . . . . . . 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 2347	. . . . . 6 ⊢ ((𝜑 ∧ Fun 𝐹) → {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧} = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴})
19		df-dm 4729	. . . . . 6 ⊢ dom 𝐹 = {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧}
20		eqid 2229	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
21	20	rnmpt 4972	. . . . . 6 ⊢ ran (𝑥 ∈ 𝑋 ↦ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴}
22	18, 19, 21	3eqtr4g 2287	. . . . 5 ⊢ ((𝜑 ∧ Fun 𝐹) → dom 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 𝐴))
23		df-fn 5321	. . . . 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 5922	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆))
26	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27		rnss 4954	. . . . . 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 5160	. . . . 5 ⊢ ran (𝑅 × 𝑆) ⊆ 𝑆
30	28, 29	sstrdi 3236	. . . 4 ⊢ ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ 𝑆)
31		df-f 5322	. . . 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 5476	. 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 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∃wrex 2509   ⊆ wss 3197  ⟨cop 3669   class class class wbr 4083   ↦ cmpt 4145   × cxp 4717  dom cdm 4719  ran crn 4720  Fun wfun 5312   Fn wfn 5313  ⟶wf 5314

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326

This theorem is referenced by:  qliftf  6775