Theorem fliftf 5700

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 109	. . . . 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 5694	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
6	5	exbidv 1797	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
7	6	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
8		rexcom4 2709	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵))
9		elisset 2700	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑆 → ∃𝑧 𝑧 = 𝐵)
10	4, 9	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑧 𝑧 = 𝐵)
11	10	biantrud 302	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 = 𝐴 ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵)))
12		19.42v 1878	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵))
13	11, 12	syl6rbbr 198	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ 𝑦 = 𝐴))
14	13	rexbidva 2434	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
15	14	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
16	8, 15	syl5bbr 193	. . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
17	7, 16	bitrd 187	. . . . . . 7 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
18	17	abbidv 2257	. . . . . 6 ⊢ ((𝜑 ∧ Fun 𝐹) → {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧} = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴})
19		df-dm 4549	. . . . . 6 ⊢ dom 𝐹 = {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧}
20		eqid 2139	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
21	20	rnmpt 4787	. . . . . 6 ⊢ ran (𝑥 ∈ 𝑋 ↦ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴}
22	18, 19, 21	3eqtr4g 2197	. . . . 5 ⊢ ((𝜑 ∧ Fun 𝐹) → dom 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 𝐴))
23		df-fn 5126	. . . . 5 ⊢ (𝐹 Fn ran (𝑥 ∈ 𝑋 ↦ 𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 𝐴)))
24	1, 22, 23	sylanbrc 413	. . . 4 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹 Fn ran (𝑥 ∈ 𝑋 ↦ 𝐴))
25	2, 3, 4	fliftrel 5693	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆))
26	25	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27		rnss 4769	. . . . . 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 4970	. . . . 5 ⊢ ran (𝑅 × 𝑆) ⊆ 𝑆
30	28, 29	sstrdi 3109	. . . 4 ⊢ ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ 𝑆)
31		df-f 5127	. . . 4 ⊢ (𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆 ↔ (𝐹 Fn ran (𝑥 ∈ 𝑋 ↦ 𝐴) ∧ ran 𝐹 ⊆ 𝑆))
32	24, 30, 31	sylanbrc 413	. . 3 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆)
33	32	ex 114	. 2 ⊢ (𝜑 → (Fun 𝐹 → 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆))
34		ffun 5275	. 2 ⊢ (𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆 → Fun 𝐹)
35	33, 34	impbid1 141	1 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2125  ∃wrex 2417   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929   ↦ cmpt 3989   × cxp 4537  dom cdm 4539  ran crn 4540  Fun wfun 5117   Fn wfn 5118  ⟶wf 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131

This theorem is referenced by:  qliftf  6514