Theorem qliftf 6717

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 6710	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	1, 5, 2	fliftf 5878	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7		df-qs 6636	. . . . 5 ⊢ (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
8		eqid 2206	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
9	8	rnmpt 4932	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
10	7, 9	eqtr4i 2230	. . . 4 ⊢ (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
11	10	a1i 9	. . 3 ⊢ (𝜑 → (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅))
12	11	feq2d 5420	. 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 1373   ∈ wcel 2177  {cab 2192  ∃wrex 2486  Vcvv 2773  ⟨cop 3638   ↦ cmpt 4110  ran crn 4681  Fun wfun 5271  ⟶wf 5273   Er wer 6627  [cec 6628   / cqs 6629

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3001  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-fv 5285  df-er 6630  df-ec 6632  df-qs 6636

This theorem is referenced by: (None)