Theorem qliftf 6468

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 6461	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	1, 5, 2	fliftf 5654	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7		df-qs 6389	. . . . 5 ⊢ (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
8		eqid 2115	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
9	8	rnmpt 4747	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
10	7, 9	eqtr4i 2138	. . . 4 ⊢ (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
11	10	a1i 9	. . 3 ⊢ (𝜑 → (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅))
12	11	feq2d 5218	. 2 ⊢ (𝜑 → (𝐹:(𝑋 / 𝑅)⟶𝑌 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)⟶𝑌))
13	6, 12	bitr4d 190	1 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:(𝑋 / 𝑅)⟶𝑌))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1314   ∈ wcel 1463  {cab 2101  ∃wrex 2391  Vcvv 2657  ⟨cop 3496   ↦ cmpt 3949  ran crn 4500  Fun wfun 5075  ⟶wf 5077   Er wer 6380  [cec 6381   / cqs 6382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-er 6383  df-ec 6385  df-qs 6389

This theorem is referenced by: (None)