Theorem qliftval 6215

Description: The value 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)
qliftval.4	⊢ (𝑥 = 𝐶 → 𝐴 = 𝐵)
qliftval.6	⊢ (𝜑 → Fun 𝐹)

Assertion

Ref	Expression
qliftval	⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)

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

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

Proof of Theorem qliftval

Step	Hyp	Ref	Expression
1		qlift.1	. 2 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2		qlift.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
3		qlift.3	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
4		qlift.4	. . 3 ⊢ (𝜑 → 𝑋 ∈ V)
5	1, 2, 3, 4	qliftlem 6207	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6		eceq1 6164	. 2 ⊢ (𝑥 = 𝐶 → [𝑥]𝑅 = [𝐶]𝑅)
7		qliftval.4	. 2 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝐵)
8		qliftval.6	. 2 ⊢ (𝜑 → Fun 𝐹)
9	1, 5, 2, 6, 7, 8	fliftval 5460	1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  Vcvv 2601  ⟨cop 3401   ↦ cmpt 3839  ran crn 4364  Fun wfun 4916  ‘cfv 4922   Er wer 6126  [cec 6127   / cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fv 4930  df-er 6129  df-ec 6131  df-qs 6135

This theorem is referenced by: (None)