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Mirrors > Home > MPE Home > Th. List > qliftval | Structured version Visualization version GIF version |
Description: The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.) |
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) |
Ref | Expression |
---|---|
qliftval | ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵) |
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 8823 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) |
6 | eceq1 8769 | . 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 7330 | 1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⟨cop 4638 ↦ cmpt 5235 ran crn 5683 Fun wfun 6547 ‘cfv 6553 Er wer 8728 [cec 8729 / cqs 8730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fv 6561 df-er 8731 df-ec 8733 df-qs 8737 |
This theorem is referenced by: orbstaval 19270 frgpupval 19736 |
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