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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.) |
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 403 | . . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝐴 = 𝐵)) |
3 | 2 | alrimivv 2027 | . 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 8102 | . 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵))) |
10 | 3, 9 | mpbird 249 | 1 ⊢ (𝜑 → Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∀wal 1654 = wceq 1656 ∈ wcel 2164 Vcvv 3414 〈cop 4405 class class class wbr 4875 ↦ cmpt 4954 ran crn 5347 Fun wfun 6121 Er wer 8011 [cec 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-er 8014 df-ec 8016 df-qs 8020 |
This theorem is referenced by: orbstafun 18101 frgpupf 18546 |
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