Theorem qliftfun 8235

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 8231	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6		eceq1 8180	. . 3 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7		qliftfun.4	. . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
8	1, 5, 2, 6, 7	fliftfun 6931	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
9	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑅 Er 𝑋)
10		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥𝑅𝑦)
11	9, 10	ercl 8153	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝑋)
12	9, 10	ercl2 8155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦 ∈ 𝑋)
13	11, 12	jca 512	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋))
14	13	ex 413	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
15	14	pm4.71rd 563	. . . . . . 7 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑅𝑦)))
16	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑅 Er 𝑋)
17		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
18	16, 17	erth 8191	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑅𝑦 ↔ [𝑥]𝑅 = [𝑦]𝑅))
19	18	pm5.32da 579	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ 𝑥𝑅𝑦) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
20	15, 19	bitrd 280	. . . . . 6 ⊢ (𝜑 → (𝑥𝑅𝑦 ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅)))
21	20	imbi1d 343	. . . . 5 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵)))
22		impexp 451	. . . . 5 ⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ [𝑥]𝑅 = [𝑦]𝑅) → 𝐴 = 𝐵) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
23	21, 22	syl6bb 288	. . . 4 ⊢ (𝜑 → ((𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵))))
24	23	2albidv 1902	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵))))
25		r2al 3167	. . 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	syl6bbr 290	. 2 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ([𝑥]𝑅 = [𝑦]𝑅 → 𝐴 = 𝐵)))
27	8, 26	bitr4d 283	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520   = wceq 1522   ∈ wcel 2080  ∀wral 3104  Vcvv 3436  ⟨cop 4480   class class class wbr 4964   ↦ cmpt 5043  ran crn 5447  Fun wfun 6222   Er wer 8139  [cec 8140   / cqs 8141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fv 6236  df-er 8142  df-ec 8144  df-qs 8148

This theorem is referenced by:  qliftfund  8236  qliftfuns  8237