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Mirrors > Home > MPE Home > Th. List > 1stcrestlem | Structured version Visualization version GIF version |
Description: Lemma for 1stcrest 22217. (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 7621 | . . . . . 6 ⊢ Ord ω | |
2 | reldom 8574 | . . . . . . . 8 ⊢ Rel ≼ | |
3 | 2 | brrelex2i 5590 | . . . . . . 7 ⊢ (𝐵 ≼ ω → ω ∈ V) |
4 | elong 6191 | . . . . . . 7 ⊢ (ω ∈ V → (ω ∈ On ↔ Ord ω)) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω)) |
6 | 1, 5 | mpbiri 261 | . . . . 5 ⊢ (𝐵 ≼ ω → ω ∈ On) |
7 | ondomen 9550 | . . . . 5 ⊢ ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card) | |
8 | 6, 7 | mpancom 688 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ dom card) |
9 | eqid 2739 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
10 | 9 | dmmptss 6083 | . . . 4 ⊢ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵 |
11 | ssnum 9552 | . . . 4 ⊢ ((𝐵 ∈ dom card ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card) | |
12 | 8, 10, 11 | sylancl 589 | . . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card) |
13 | funmpt 6388 | . . . 4 ⊢ Fun (𝑥 ∈ 𝐵 ↦ 𝐶) | |
14 | funforn 6610 | . . . 4 ⊢ (Fun (𝑥 ∈ 𝐵 ↦ 𝐶) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
15 | 13, 14 | mpbi 233 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶) |
16 | fodomnum 9570 | . . 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 8583 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
19 | ssdomg 8614 | . . . 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 8621 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵 ∧ 𝐵 ≼ ω) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | |
22 | 20, 21 | mpancom 688 | . 2 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
23 | domtr 8621 | . 2 ⊢ ((ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) | |
24 | 17, 22, 23 | syl2anc 587 | 1 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2114 Vcvv 3400 ⊆ wss 3853 class class class wbr 5040 ↦ cmpt 5120 dom cdm 5535 ran crn 5536 Ord word 6182 Oncon0 6183 Fun wfun 6344 –onto→wfo 6348 ωcom 7612 ≼ cdom 8566 cardccrd 9450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-er 8333 df-map 8452 df-en 8569 df-dom 8570 df-card 9454 df-acn 9457 |
This theorem is referenced by: 1stcrest 22217 2ndcrest 22218 lly1stc 22260 abrexct 30639 ldgenpisyslem1 31714 saliuncl 43446 meadjiun 43587 |
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