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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnurnd | Structured version Visualization version GIF version |
Description: Minimal universes contain ranges of functions from an element of the universe to the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
mnurnd.1 | ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} |
mnurnd.2 | ⊢ (𝜑 → 𝑈 ∈ 𝑀) |
mnurnd.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
mnurnd.4 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑈) |
Ref | Expression |
---|---|
mnurnd | ⊢ (𝜑 → ran 𝐹 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnurnd.1 | . 2 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))} | |
2 | mnurnd.2 | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀) | |
3 | mnurnd.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 3 | elexd 3467 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | 4 | iftrued 4498 | . . 3 ⊢ (𝜑 → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴) |
6 | 5, 3 | eqeltrd 2834 | . 2 ⊢ (𝜑 → if(𝐴 ∈ V, 𝐴, ∅) ∈ 𝑈) |
7 | mnurnd.4 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑈) | |
8 | 5 | feq2d 6658 | . . 3 ⊢ (𝜑 → (𝐹:if(𝐴 ∈ V, 𝐴, ∅)⟶𝑈 ↔ 𝐹:𝐴⟶𝑈)) |
9 | 7, 8 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐹:if(𝐴 ∈ V, 𝐴, ∅)⟶𝑈) |
10 | 0ex 5268 | . . 3 ⊢ ∅ ∈ V | |
11 | 10 | elimel 4559 | . 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) ∈ V |
12 | 1, 2, 6, 9, 11 | mnurndlem2 42654 | 1 ⊢ (𝜑 → ran 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3061 ∃wrex 3070 Vcvv 3447 ⊆ wss 3914 ∅c0 4286 ifcif 4490 𝒫 cpw 4564 ∪ cuni 4869 ran crn 5638 ⟶wf 6496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-reg 9536 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-eprel 5541 df-fr 5592 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 |
This theorem is referenced by: mnugrud 42656 |
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