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Mirrors > Home > MPE Home > Th. List > qliftfund | Structured version Visualization version GIF version |
Description: The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (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) |
qliftfun.4 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
qliftfund.6 | ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
qliftfund | ⊢ (𝜑 → Fun 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qliftfund.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝐴 = 𝐵) | |
2 | 1 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝐴 = 𝐵)) |
3 | 2 | alrimivv 1926 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)) |
4 | qlift.1 | . . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) | |
5 | qlift.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) | |
6 | qlift.3 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
7 | qlift.4 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
8 | qliftfun.4 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
9 | 4, 5, 6, 7, 8 | qliftfun 8841 | . 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵))) |
10 | 3, 9 | mpbird 257 | 1 ⊢ (𝜑 → Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 Fun wfun 6557 Er wer 8741 [cec 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-er 8744 df-ec 8746 df-qs 8750 |
This theorem is referenced by: orbstafun 19342 frgpupf 19806 |
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