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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 1928 | . 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 8842 | . 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 1538 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 Fun wfun 6555 Er wer 8742 [cec 8743 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-er 8745 df-ec 8747 df-qs 8751 | 
| This theorem is referenced by: orbstafun 19329 frgpupf 19791 | 
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