Theorem qliftfun 6829

Description: The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
qlift.1	⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
qlift.3	⊢ (𝜑 → 𝑅 Er 𝑋)
qlift.4	⊢ (𝜑 → 𝑋 ∈ V)
qliftfun.4	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
qliftfun	⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦,𝜑   𝑥,𝑅,𝑦   𝑦,𝐹   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

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

Proof of Theorem qliftfun

Step	Hyp	Ref	Expression
1		qlift.1	. . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2		qlift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
3		qlift.3	. . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋)
4		qlift.4	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
5	1, 2, 3, 4	qliftlem 6825	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6		eceq1 6780	. . 3 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7		qliftfun.4	. . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
8	1, 5, 2, 6, 7	fliftfun 5947	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
9	3	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑅 Er 𝑋)
10		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
11	9, 10	ercl 6756	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝑋)
12	9, 10	ercl2 6758	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝑋)
13	11, 12	jca 306	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
14	13	ex 115	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
15	14	pm4.71rd 394	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑅𝑦)))
16	3	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑅 Er 𝑋)
17		simprl 531	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
18	16, 17	erth 6791	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
19	18	pm5.32da 452	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑅𝑦) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
20	15, 19	bitrd 188	. . . . . 6 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
21	20	imbi1d 231	. . . . 5 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵)))
22		impexp 263	. . . . 5 ⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
23	21, 22	bitrdi 196	. . . 4 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵))))
24	23	2albidv 1915	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵))))
25		r2al 2552	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
26	24, 25	bitr4di 198	. 2 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
27	8, 26	bitr4d 191	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803  ⟨cop 3676   class class class wbr 4093   ↦ cmpt 4155  ran crn 4732  Fun wfun 5327   Er wer 6742  [cec 6743   / cqs 6744

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-er 6745  df-ec 6747  df-qs 6751

This theorem is referenced by:  qliftfund  6830  qliftfuns  6831