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Mirrors > Home > MPE Home > Th. List > 1stcrestlem | Structured version Visualization version GIF version |
Description: Lemma for 1stcrest 21256. (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 7074 | . . . . . 6 ⊢ Ord ω | |
2 | reldom 7961 | . . . . . . . 8 ⊢ Rel ≼ | |
3 | 2 | brrelex2i 5159 | . . . . . . 7 ⊢ (𝐵 ≼ ω → ω ∈ V) |
4 | elong 5731 | . . . . . . 7 ⊢ (ω ∈ V → (ω ∈ On ↔ Ord ω)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω)) |
6 | 1, 5 | mpbiri 248 | . . . . 5 ⊢ (𝐵 ≼ ω → ω ∈ On) |
7 | ondomen 8860 | . . . . 5 ⊢ ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card) | |
8 | 6, 7 | mpancom 703 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ dom card) |
9 | eqid 2622 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
10 | 9 | dmmptss 5631 | . . . 4 ⊢ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵 |
11 | ssnum 8862 | . . . 4 ⊢ ((𝐵 ∈ dom card ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card) | |
12 | 8, 10, 11 | sylancl 694 | . . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card) |
13 | funmpt 5926 | . . . 4 ⊢ Fun (𝑥 ∈ 𝐵 ↦ 𝐶) | |
14 | funforn 6122 | . . . 4 ⊢ (Fun (𝑥 ∈ 𝐵 ↦ 𝐶) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
15 | 13, 14 | mpbi 220 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶) |
16 | fodomnum 8880 | . . 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 | 2 | brrelexi 5158 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) |
19 | ssdomg 8001 | . . . 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 8009 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵 ∧ 𝐵 ≼ ω) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | |
22 | 20, 21 | mpancom 703 | . 2 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
23 | domtr 8009 | . 2 ⊢ ((ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | |
24 | 17, 22, 23 | syl2anc 693 | 1 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 Ord word 5722 Oncon0 5723 Fun wfun 5882 –onto→wfo 5886 ωcom 7065 ≼ cdom 7953 cardccrd 8761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-card 8765 df-acn 8768 |
This theorem is referenced by: 1stcrest 21256 2ndcrest 21257 lly1stc 21299 abrexct 29494 ldgenpisyslem1 30226 saliuncl 40542 meadjiun 40683 |
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