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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  ontowfo 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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