Theorem qliftfun 6851

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 6847	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6		eceq1 6802	. . 3 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7		qliftfun.4	. . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
8	1, 5, 2, 6, 7	fliftfun 5969	. 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 6778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝑋)
12	9, 10	ercl2 6780	. . . . . . . . . 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 6813	. . . . . . . 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 1916	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵))))
25		r2al 2561	. . 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 2203  ∀wral 2520  Vcvv 2813  ⟨cop 3692   class class class wbr 4109   ↦ cmpt 4171  ran crn 4750  Fun wfun 5346   Er wer 6764  [cec 6765   / cqs 6766

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-er 6767  df-ec 6769  df-qs 6773

This theorem is referenced by:  qliftfund  6852  qliftfuns  6853