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Mirrors > Home > ILE Home > Th. List > qliftval | GIF version |
Description: The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
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 6386 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) |
6 | eceq1 6343 | . 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 5595 | 1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 Vcvv 2622 〈cop 3455 ↦ cmpt 3907 ran crn 4455 Fun wfun 5024 ‘cfv 5030 Er wer 6305 [cec 6306 / cqs 6307 |
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 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2624 df-sbc 2844 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-mpt 3909 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fv 5038 df-er 6308 df-ec 6310 df-qs 6314 |
This theorem is referenced by: (None) |
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