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Mirrors > Home > MPE Home > Th. List > hartogslem2 | Structured version Visualization version GIF version |
Description: Lemma for hartogs 9559. (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 9557 | . . 3 ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})) |
4 | 3 | simp3i 1139 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}) |
5 | 3 | simp2i 1138 | . . . 4 ⊢ Fun 𝐹 |
6 | 3 | simp1i 1137 | . . . . 5 ⊢ dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) |
7 | sqxpexg 7751 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
8 | 7 | pwexd 5373 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V) |
9 | ssexg 5317 | . . . . 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 7225 | . . . 4 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
12 | 5, 10, 11 | sylancr 586 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
13 | rnexg 7904 | . . 3 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐹 ∈ V) |
15 | 4, 14 | eqeltrrd 2829 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 {crab 3427 Vcvv 3469 ∖ cdif 3941 ⊆ wss 3944 𝒫 cpw 4598 class class class wbr 5142 {copab 5204 I cid 5569 E cep 5575 We wwe 5626 × cxp 5670 dom cdm 5672 ran crn 5673 ↾ cres 5674 Oncon0 6363 Fun wfun 6536 ‘cfv 6542 ≼ cdom 8953 OrdIsocoi 9524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-en 8956 df-dom 8957 df-oi 9525 |
This theorem is referenced by: hartogs 9559 |
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