Theorem fliftf 7182

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 485	. . . . 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 7176	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
6	5	exbidv 1928	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
7	6	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵)))
8		rexcom4 3182	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵))
9		19.42v 1961	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵))
10		elisset 2822	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑆 → ∃𝑧 𝑧 = 𝐵)
11	4, 10	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑧 𝑧 = 𝐵)
12	11	biantrud 532	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 = 𝐴 ↔ (𝑦 = 𝐴 ∧ ∃𝑧 𝑧 = 𝐵)))
13	9, 12	bitr4id 290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ 𝑦 = 𝐴))
14	13	rexbidva 3227	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
15	14	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑥 ∈ 𝑋 ∃𝑧(𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
16	8, 15	bitr3id 285	. . . . . . . 8 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑧∃𝑥 ∈ 𝑋 (𝑦 = 𝐴 ∧ 𝑧 = 𝐵) ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
17	7, 16	bitrd 278	. . . . . . 7 ⊢ ((𝜑 ∧ Fun 𝐹) → (∃𝑧 𝑦𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴))
18	17	abbidv 2809	. . . . . 6 ⊢ ((𝜑 ∧ Fun 𝐹) → {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧} = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴})
19		df-dm 5600	. . . . . 6 ⊢ dom 𝐹 = {𝑦 ∣ ∃𝑧 𝑦𝐹𝑧}
20		eqid 2740	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
21	20	rnmpt 5863	. . . . . 6 ⊢ ran (𝑥 ∈ 𝑋 ↦ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = 𝐴}
22	18, 19, 21	3eqtr4g 2805	. . . . 5 ⊢ ((𝜑 ∧ Fun 𝐹) → dom 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 𝐴))
23		df-fn 6435	. . . . 5 ⊢ (𝐹 Fn ran (𝑥 ∈ 𝑋 ↦ 𝐴) ↔ (Fun 𝐹 ∧ dom 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 𝐴)))
24	1, 22, 23	sylanbrc 583	. . . 4 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹 Fn ran (𝑥 ∈ 𝑋 ↦ 𝐴))
25	2, 3, 4	fliftrel 7175	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆))
26	25	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹 ⊆ (𝑅 × 𝑆))
27		rnss 5847	. . . . . 6 ⊢ (𝐹 ⊆ (𝑅 × 𝑆) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
28	26, 27	syl 17	. . . . 5 ⊢ ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ ran (𝑅 × 𝑆))
29		rnxpss 6074	. . . . 5 ⊢ ran (𝑅 × 𝑆) ⊆ 𝑆
30	28, 29	sstrdi 3938	. . . 4 ⊢ ((𝜑 ∧ Fun 𝐹) → ran 𝐹 ⊆ 𝑆)
31		df-f 6436	. . . 4 ⊢ (𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆 ↔ (𝐹 Fn ran (𝑥 ∈ 𝑋 ↦ 𝐴) ∧ ran 𝐹 ⊆ 𝑆))
32	24, 30, 31	sylanbrc 583	. . 3 ⊢ ((𝜑 ∧ Fun 𝐹) → 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆)
33	32	ex 413	. 2 ⊢ (𝜑 → (Fun 𝐹 → 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆))
34		ffun 6601	. 2 ⊢ (𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆 → Fun 𝐹)
35	33, 34	impbid1 224	1 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542  ∃wex 1786   ∈ wcel 2110  {cab 2717  ∃wrex 3067   ⊆ wss 3892  ⟨cop 4573   class class class wbr 5079   ↦ cmpt 5162   × cxp 5588  dom cdm 5590  ran crn 5591  Fun wfun 6426   Fn wfn 6427  ⟶wf 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-fun 6434  df-fn 6435  df-f 6436

This theorem is referenced by:  qliftf  8577  cygznlem2a  20773  pi1xfrf  24214  pi1cof  24220