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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 449 | . . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → 𝐴 = 𝐵)) |
3 | 2 | alrimivv 1843 | . 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 7719 | . 2 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵))) |
10 | 3, 9 | mpbird 246 | 1 ⊢ (𝜑 → Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 Fun wfun 5798 Er wer 7626 [cec 7627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-er 7629 df-ec 7631 df-qs 7635 |
This theorem is referenced by: orbstafun 17567 frgpupf 18009 |
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