Theorem qliftf 7987

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 7980	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	1, 5, 2	fliftf 6708	. 2 ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7		df-qs 7902	. . . . 5 ⊢ (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
8		eqid 2771	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
9	8	rnmpt 5509	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝑋 𝑦 = [𝑥]𝑅}
10	7, 9	eqtr4i 2796	. . . 4 ⊢ (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅)
11	10	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 / 𝑅) = ran (𝑥 ∈ 𝑋 ↦ [𝑥]𝑅))
12	11	feq2d 6171	. 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 382   = wceq 1631   ∈ wcel 2145  {cab 2757  ∃wrex 3062  Vcvv 3351  ⟨cop 4322   ↦ cmpt 4863  ran crn 5250  Fun wfun 6025  ⟶wf 6027   Er wer 7893  [cec 7894   / cqs 7895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-er 7896  df-ec 7898  df-qs 7902

This theorem is referenced by:  orbsta  17953  frgpupf  18393