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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 3511 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | 4 | iftrued 4468 | . . 3 ⊢ (𝜑 → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴) |
6 | 5, 3 | eqeltrd 2912 | . 2 ⊢ (𝜑 → if(𝐴 ∈ V, 𝐴, ∅) ∈ 𝑈) |
7 | mnurnd.4 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑈) | |
8 | 5 | feq2d 6493 | . . 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 5204 | . . 3 ⊢ ∅ ∈ V | |
11 | 10 | elimel 4527 | . 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) ∈ V |
12 | 1, 2, 6, 9, 11 | mnurndlem2 40693 | 1 ⊢ (𝜑 → ran 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1534 = wceq 1536 ∈ wcel 2113 {cab 2798 ∀wral 3137 ∃wrex 3138 Vcvv 3491 ⊆ wss 3929 ∅c0 4284 ifcif 4460 𝒫 cpw 4532 ∪ cuni 4831 ran crn 5549 ⟶wf 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-reg 9049 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-eprel 5458 df-fr 5507 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 |
This theorem is referenced by: mnugrud 40695 |
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