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Mirrors > Home > MPE Home > Th. List > hartogslem2 | Structured version Visualization version GIF version |
Description: Lemma for hartogs 9541. (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 9539 | . . 3 ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})) |
4 | 3 | simp3i 1138 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) |
5 | 3 | simp2i 1137 | . . . 4 ⊢ Fun 𝐹 |
6 | 3 | simp1i 1136 | . . . . 5 ⊢ dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) |
7 | sqxpexg 7739 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
8 | 7 | pwexd 5370 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V) |
9 | ssexg 5316 | . . . . 5 ⊢ ((dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ 𝒫 (𝐴 × 𝐴) ∈ V) → dom 𝐹 ∈ V) | |
10 | 6, 8, 9 | sylancr 586 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ V) |
11 | funex 7216 | . . . 4 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
12 | 5, 10, 11 | sylancr 586 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
13 | rnexg 7892 | . . 3 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 ∈ V) |
15 | 4, 14 | eqeltrrd 2828 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 {crab 3426 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 𝒫 cpw 4597 class class class wbr 5141 {copab 5203 I cid 5566 E cep 5572 We wwe 5623 × cxp 5667 dom cdm 5669 ran crn 5670 ↾ cres 5671 Oncon0 6358 Fun wfun 6531 ‘cfv 6537 ≼ cdom 8939 OrdIsocoi 9506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-en 8942 df-dom 8943 df-oi 9507 |
This theorem is referenced by: hartogs 9541 |
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