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| Mirrors > Home > MPE Home > Th. List > 1stcrestlem | Structured version Visualization version GIF version | ||
| Description: Lemma for 1stcrest 23461. (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 7897 | . . . . . 6 ⊢ Ord ω | |
| 2 | reldom 8991 | . . . . . . . 8 ⊢ Rel ≼ | |
| 3 | 2 | brrelex2i 5742 | . . . . . . 7 ⊢ (𝐵 ≼ ω → ω ∈ V) |
| 4 | elong 6392 | . . . . . . 7 ⊢ (ω ∈ V → (ω ∈ On ↔ Ord ω)) | |
| 5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω)) |
| 6 | 1, 5 | mpbiri 258 | . . . . 5 ⊢ (𝐵 ≼ ω → ω ∈ On) |
| 7 | ondomen 10077 | . . . . 5 ⊢ ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card) | |
| 8 | 6, 7 | mpancom 688 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ dom card) |
| 9 | eqid 2737 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
| 10 | 9 | dmmptss 6261 | . . . 4 ⊢ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵 |
| 11 | ssnum 10079 | . . . 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 6604 | . . . 4 ⊢ Fun (𝑥 ∈ 𝐵 ↦ 𝐶) | |
| 14 | funforn 6827 | . . . 4 ⊢ (Fun (𝑥 ∈ 𝐵 ↦ 𝐶) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
| 15 | 13, 14 | mpbi 230 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶) |
| 16 | fodomnum 10097 | . . 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 9004 | . . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
| 19 | ssdomg 9040 | . . . 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 9047 | . . 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 9047 | . 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 206 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 Ord word 6383 Oncon0 6384 Fun wfun 6555 –onto→wfo 6559 ωcom 7887 ≼ cdom 8983 cardccrd 9975 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-card 9979 df-acn 9982 |
| This theorem is referenced by: 1stcrest 23461 2ndcrest 23462 lly1stc 23504 abrexct 32728 ldgenpisyslem1 34164 meadjiun 46481 |
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