Theorem qliftf 6622

Description: The domain and codomain 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 6615	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	1, 5, 2	fliftf 5802	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7		df-qs 6543	. . . . 5 ⊢ (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
8		eqid 2177	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
9	8	rnmpt 4877	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
10	7, 9	eqtr4i 2201	. . . 4 ⊢ (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
11	10	a1i 9	. . 3 ⊢ (𝜑 → (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅))
12	11	feq2d 5355	. 2 ⊢ (𝜑 → (𝐹:(𝑋 / 𝑅)⟶𝑌 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)⟶𝑌))
13	6, 12	bitr4d 191	1 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:(𝑋 / 𝑅)⟶𝑌))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  ∃wrex 2456  Vcvv 2739  ⟨cop 3597   ↦ cmpt 4066  ran crn 4629  Fun wfun 5212  ⟶wf 5214   Er wer 6534  [cec 6535   / cqs 6536

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-er 6537  df-ec 6539  df-qs 6543

This theorem is referenced by: (None)