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Theorem 1stcrestlem 21978
 Description: Lemma for 1stcrest 21979. (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 7580 . . . . . 6 Ord ω
2 reldom 8507 . . . . . . . 8 Rel ≼
32brrelex2i 5607 . . . . . . 7 (𝐵 ≼ ω → ω ∈ V)
4 elong 6196 . . . . . . 7 (ω ∈ V → (ω ∈ On ↔ Ord ω))
53, 4syl 17 . . . . . 6 (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω))
61, 5mpbiri 259 . . . . 5 (𝐵 ≼ ω → ω ∈ On)
7 ondomen 9455 . . . . 5 ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card)
86, 7mpancom 684 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ dom card)
9 eqid 2825 . . . . 5 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
109dmmptss 6092 . . . 4 dom (𝑥𝐵𝐶) ⊆ 𝐵
11 ssnum 9457 . . . 4 ((𝐵 ∈ dom card ∧ dom (𝑥𝐵𝐶) ⊆ 𝐵) → dom (𝑥𝐵𝐶) ∈ dom card)
128, 10, 11sylancl 586 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ∈ dom card)
13 funmpt 6389 . . . 4 Fun (𝑥𝐵𝐶)
14 funforn 6593 . . . 4 (Fun (𝑥𝐵𝐶) ↔ (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶))
1513, 14mpbi 231 . . 3 (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶)
16 fodomnum 9475 . . 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))
18 ctex 8516 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ V)
19 ssdomg 8547 . . . 4 (𝐵 ∈ V → (dom (𝑥𝐵𝐶) ⊆ 𝐵 → dom (𝑥𝐵𝐶) ≼ 𝐵))
2018, 10, 19mpisyl 21 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ 𝐵)
21 domtr 8554 . . 3 ((dom (𝑥𝐵𝐶) ≼ 𝐵𝐵 ≼ ω) → dom (𝑥𝐵𝐶) ≼ ω)
2220, 21mpancom 684 . 2 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ ω)
23 domtr 8554 . 2 ((ran (𝑥𝐵𝐶) ≼ dom (𝑥𝐵𝐶) ∧ dom (𝑥𝐵𝐶) ≼ ω) → ran (𝑥𝐵𝐶) ≼ ω)
2417, 22, 23syl2anc 584 1 (𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2107  Vcvv 3499   ⊆ wss 3939   class class class wbr 5062   ↦ cmpt 5142  dom cdm 5553  ran crn 5554  Ord word 6187  Oncon0 6188  Fun wfun 6345  –onto→wfo 6349  ωcom 7571   ≼ cdom 8499  cardccrd 9356 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-card 9360  df-acn 9363 This theorem is referenced by:  1stcrest  21979  2ndcrest  21980  lly1stc  22022  abrexct  30367  ldgenpisyslem1  31310  saliuncl  42475  meadjiun  42616
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