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Mirrors > Home > MPE Home > Th. List > hartogslem2 | Structured version Visualization version GIF version |
Description: Lemma for hartogs 8797. (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 8795 | . . 3 ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})) |
4 | 3 | simp3i 1121 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) |
5 | 3 | simp2i 1120 | . . . 4 ⊢ Fun 𝐹 |
6 | 3 | simp1i 1119 | . . . . 5 ⊢ dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) |
7 | sqxpexg 7288 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
8 | 7 | pwexd 5127 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V) |
9 | ssexg 5077 | . . . . 5 ⊢ ((dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ 𝒫 (𝐴 × 𝐴) ∈ V) → dom 𝐹 ∈ V) | |
10 | 6, 8, 9 | sylancr 578 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ V) |
11 | funex 6802 | . . . 4 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
12 | 5, 10, 11 | sylancr 578 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
13 | rnexg 7423 | . . 3 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 ∈ V) |
15 | 4, 14 | eqeltrrd 2861 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∃wrex 3083 {crab 3086 Vcvv 3409 ∖ cdif 3820 ⊆ wss 3823 𝒫 cpw 4416 class class class wbr 4923 {copab 4985 I cid 5305 E cep 5310 We wwe 5359 × cxp 5399 dom cdm 5401 ran crn 5402 ↾ cres 5403 Oncon0 6023 Fun wfun 6176 ‘cfv 6182 ≼ cdom 8298 OrdIsocoi 8762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-wrecs 7744 df-recs 7806 df-en 8301 df-dom 8302 df-oi 8763 |
This theorem is referenced by: hartogs 8797 |
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