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Mirrors > Home > MPE Home > Th. List > hartogslem2 | Structured version Visualization version GIF version |
Description: Lemma for hartogs 8332. (Contributed by Mario Carneiro, 14-Jan-2013.) |
Ref | Expression |
---|---|
hartogslem.2 | ⊢ 𝐹 = {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} |
hartogslem.3 | ⊢ 𝑅 = {〈𝑠, 𝑡〉 ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)} |
Ref | Expression |
---|---|
hartogslem2 | ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hartogslem.2 | . . . 4 ⊢ 𝐹 = {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} | |
2 | hartogslem.3 | . . . 4 ⊢ 𝑅 = {〈𝑠, 𝑡〉 ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)} | |
3 | 1, 2 | hartogslem1 8330 | . . 3 ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})) |
4 | 3 | simp3i 1065 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) |
5 | 3 | simp2i 1064 | . . . 4 ⊢ Fun 𝐹 |
6 | 3 | simp1i 1063 | . . . . 5 ⊢ dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) |
7 | sqxpexg 6861 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
8 | pwexg 4776 | . . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → 𝒫 (𝐴 × 𝐴) ∈ V) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V) |
10 | ssexg 4732 | . . . . 5 ⊢ ((dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ 𝒫 (𝐴 × 𝐴) ∈ V) → dom 𝐹 ∈ V) | |
11 | 6, 9, 10 | sylancr 694 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ V) |
12 | funex 6387 | . . . 4 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
13 | 5, 11, 12 | sylancr 694 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
14 | rnexg 6990 | . . 3 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 ∈ V) |
16 | 4, 15 | eqeltrrd 2689 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 {copab 4642 E cep 4947 I cid 4948 We wwe 4996 × cxp 5036 dom cdm 5038 ran crn 5039 ↾ cres 5040 Oncon0 5640 Fun wfun 5798 ‘cfv 5804 ≼ cdom 7839 OrdIsocoi 8297 |
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-rep 4699 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-3or 1032 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-reu 2903 df-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-wrecs 7294 df-recs 7355 df-en 7842 df-dom 7843 df-oi 8298 |
This theorem is referenced by: hartogs 8332 |
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