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Mirrors > Home > MPE Home > Th. List > 1stcrestlem | Structured version Visualization version GIF version |
Description: Lemma for 1stcrest 21989. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
1stcrestlem | ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 7578 | . . . . . 6 ⊢ Ord ω | |
2 | reldom 8503 | . . . . . . . 8 ⊢ Rel ≼ | |
3 | 2 | brrelex2i 5602 | . . . . . . 7 ⊢ (𝐵 ≼ ω → ω ∈ V) |
4 | elong 6192 | . . . . . . 7 ⊢ (ω ∈ V → (ω ∈ On ↔ Ord ω)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω)) |
6 | 1, 5 | mpbiri 259 | . . . . 5 ⊢ (𝐵 ≼ ω → ω ∈ On) |
7 | ondomen 9451 | . . . . 5 ⊢ ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card) | |
8 | 6, 7 | mpancom 684 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ dom card) |
9 | eqid 2818 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
10 | 9 | dmmptss 6088 | . . . 4 ⊢ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵 |
11 | ssnum 9453 | . . . 4 ⊢ ((𝐵 ∈ dom card ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card) | |
12 | 8, 10, 11 | sylancl 586 | . . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card) |
13 | funmpt 6386 | . . . 4 ⊢ Fun (𝑥 ∈ 𝐵 ↦ 𝐶) | |
14 | funforn 6590 | . . . 4 ⊢ (Fun (𝑥 ∈ 𝐵 ↦ 𝐶) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
15 | 13, 14 | mpbi 231 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶) |
16 | fodomnum 9471 | . . 3 ⊢ (dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card → ((𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶))) | |
17 | 12, 15, 16 | mpisyl 21 | . 2 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶)) |
18 | ctex 8512 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
19 | ssdomg 8543 | . . . 4 ⊢ (𝐵 ∈ V → (dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵 → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵)) | |
20 | 18, 10, 19 | mpisyl 21 | . . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵) |
21 | domtr 8550 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵 ∧ 𝐵 ≼ ω) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | |
22 | 20, 21 | mpancom 684 | . 2 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
23 | domtr 8550 | . 2 ⊢ ((ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | |
24 | 17, 22, 23 | syl2anc 584 | 1 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ran crn 5549 Ord word 6183 Oncon0 6184 Fun wfun 6342 –onto→wfo 6346 ωcom 7569 ≼ cdom 8495 cardccrd 9352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-card 9356 df-acn 9359 |
This theorem is referenced by: 1stcrest 21989 2ndcrest 21990 lly1stc 22032 abrexct 30378 ldgenpisyslem1 31321 saliuncl 42484 meadjiun 42625 |
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