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| Mirrors > Home > MPE Home > Th. List > hartogslem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for hartogs 9494. (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 9492 | . . 3 ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})) |
| 4 | 3 | simp3i 1157 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) |
| 5 | 3 | simp2i 1156 | . . . 4 ⊢ Fun 𝐹 |
| 6 | 3 | simp1i 1155 | . . . . 5 ⊢ dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) |
| 7 | sqxpexg 7742 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
| 8 | 7 | pwexd 5340 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V) |
| 9 | ssexg 5283 | . . . . 5 ⊢ ((dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ 𝒫 (𝐴 × 𝐴) ∈ V) → dom 𝐹 ∈ V) | |
| 10 | 6, 8, 9 | sylancr 598 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ V) |
| 11 | funex 7207 | . . . 4 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
| 12 | 5, 10, 11 | sylancr 598 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
| 13 | rnexg 7887 | . . 3 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
| 14 | 12, 13 | syl 18 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 ∈ V) |
| 15 | 4, 14 | eqeltrrd 2866 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∃wrex 3089 {crab 3417 Vcvv 3457 ∖ cdif 3904 ⊆ wss 3907 𝒫 cpw 4558 class class class wbr 5104 {copab 5166 I cid 5545 E cep 5550 We wwe 5603 × cxp 5649 dom cdm 5651 ran crn 5652 ↾ cres 5653 Oncon0 6349 Fun wfun 6519 ‘cfv 6525 ≼ cdom 8929 OrdIsocoi 9459 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-en 8932 df-dom 8933 df-oi 9460 |
| This theorem is referenced by: hartogs 9494 |
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