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Mirrors > Home > ILE Home > Th. List > qliftfund | 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.) |
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 115 | . . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝐴 = 𝐵)) |
3 | 2 | alrimivv 1875 | . 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 6617 | . 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵))) |
10 | 3, 9 | mpbird 167 | 1 ⊢ (𝜑 → Fun 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1351 = wceq 1353 ∈ wcel 2148 Vcvv 2738 ⟨cop 3596 class class class wbr 4004 ↦ cmpt 4065 ran crn 4628 Fun wfun 5211 Er wer 6532 [cec 6533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fv 5225 df-er 6535 df-ec 6537 df-qs 6541 |
This theorem is referenced by: (None) |
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