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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 3514 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | 4 | iftrued 4475 | . . 3 ⊢ (𝜑 → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴) |
6 | 5, 3 | eqeltrd 2913 | . 2 ⊢ (𝜑 → if(𝐴 ∈ V, 𝐴, ∅) ∈ 𝑈) |
7 | mnurnd.4 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑈) | |
8 | 5 | feq2d 6500 | . . 3 ⊢ (𝜑 → (𝐹:if(𝐴 ∈ V, 𝐴, ∅)⟶𝑈 ↔ 𝐹:𝐴⟶𝑈)) |
9 | 7, 8 | mpbird 259 | . 2 ⊢ (𝜑 → 𝐹:if(𝐴 ∈ V, 𝐴, ∅)⟶𝑈) |
10 | 0ex 5211 | . . 3 ⊢ ∅ ∈ V | |
11 | 10 | elimel 4534 | . 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) ∈ V |
12 | 1, 2, 6, 9, 11 | mnurndlem2 40667 | 1 ⊢ (𝜑 → ran 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 {cab 2799 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 ifcif 4467 𝒫 cpw 4539 ∪ cuni 4838 ran crn 5556 ⟶wf 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-reg 9056 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-eprel 5465 df-fr 5514 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 |
This theorem is referenced by: mnugrud 40669 |
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