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Theorem 1stcrestlem 21236
 Description: Lemma for 1stcrest 21237. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
1stcrestlem (𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)
Distinct variable group:   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem 1stcrestlem
StepHypRef Expression
1 ordom 7059 . . . . . 6 Ord ω
2 reldom 7946 . . . . . . . 8 Rel ≼
32brrelex2i 5149 . . . . . . 7 (𝐵 ≼ ω → ω ∈ V)
4 elong 5719 . . . . . . 7 (ω ∈ V → (ω ∈ On ↔ Ord ω))
53, 4syl 17 . . . . . 6 (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω))
61, 5mpbiri 248 . . . . 5 (𝐵 ≼ ω → ω ∈ On)
7 ondomen 8845 . . . . 5 ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card)
86, 7mpancom 702 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ dom card)
9 eqid 2620 . . . . 5 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
109dmmptss 5619 . . . 4 dom (𝑥𝐵𝐶) ⊆ 𝐵
11 ssnum 8847 . . . 4 ((𝐵 ∈ dom card ∧ dom (𝑥𝐵𝐶) ⊆ 𝐵) → dom (𝑥𝐵𝐶) ∈ dom card)
128, 10, 11sylancl 693 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ∈ dom card)
13 funmpt 5914 . . . 4 Fun (𝑥𝐵𝐶)
14 funforn 6109 . . . 4 (Fun (𝑥𝐵𝐶) ↔ (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶))
1513, 14mpbi 220 . . 3 (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶)
16 fodomnum 8865 . . 3 (dom (𝑥𝐵𝐶) ∈ dom card → ((𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶) → ran (𝑥𝐵𝐶) ≼ dom (𝑥𝐵𝐶)))
1712, 15, 16mpisyl 21 . 2 (𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ dom (𝑥𝐵𝐶))
182brrelexi 5148 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ V)
19 ssdomg 7986 . . . 4 (𝐵 ∈ V → (dom (𝑥𝐵𝐶) ⊆ 𝐵 → dom (𝑥𝐵𝐶) ≼ 𝐵))
2018, 10, 19mpisyl 21 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ 𝐵)
21 domtr 7994 . . 3 ((dom (𝑥𝐵𝐶) ≼ 𝐵𝐵 ≼ ω) → dom (𝑥𝐵𝐶) ≼ ω)
2220, 21mpancom 702 . 2 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ ω)
23 domtr 7994 . 2 ((ran (𝑥𝐵𝐶) ≼ dom (𝑥𝐵𝐶) ∧ dom (𝑥𝐵𝐶) ≼ ω) → ran (𝑥𝐵𝐶) ≼ ω)
2417, 22, 23syl2anc 692 1 (𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1988  Vcvv 3195   ⊆ wss 3567   class class class wbr 4644   ↦ cmpt 4720  dom cdm 5104  ran crn 5105  Ord word 5710  Oncon0 5711  Fun wfun 5870  –onto→wfo 5874  ωcom 7050   ≼ cdom 7938  cardccrd 8746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-card 8750  df-acn 8753 This theorem is referenced by:  1stcrest  21237  2ndcrest  21238  lly1stc  21280  abrexct  29468  ldgenpisyslem1  30200  saliuncl  40305  meadjiun  40446
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