Theorem qliftf 7878

Description: The domain and range of the function 𝐹. (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)

Assertion

Ref	Expression
qliftf	⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:(𝑋 / 𝑅)⟶𝑌))

Distinct variable groups:   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌

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

Proof of Theorem qliftf

Dummy variable 𝑦 is distinct from all other variables.

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 7871	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	1, 5, 2	fliftf 6605	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7		df-qs 7793	. . . . 5 ⊢ (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
8		eqid 2651	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
9	8	rnmpt 5403	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
10	7, 9	eqtr4i 2676	. . . 4 ⊢ (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
11	10	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅))
12	11	feq2d 6069	. 2 ⊢ (𝜑 → (𝐹:(𝑋 / 𝑅)⟶𝑌 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)⟶𝑌))
13	6, 12	bitr4d 271	1 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:(𝑋 / 𝑅)⟶𝑌))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  ∃wrex 2942  Vcvv 3231  ⟨cop 4216   ↦ cmpt 4762  ran crn 5144  Fun wfun 5920  ⟶wf 5922   Er wer 7784  [cec 7785   / cqs 7786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-er 7787  df-ec 7789  df-qs 7793

This theorem is referenced by:  orbsta  17792  frgpupf  18232