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Theorem 1stcrestlem 21476
Description: Lemma for 1stcrest 21477. (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 7221 . . . . . 6 Ord ω
2 reldom 8115 . . . . . . . 8 Rel ≼
32brrelex2i 5299 . . . . . . 7 (𝐵 ≼ ω → ω ∈ V)
4 elong 5874 . . . . . . 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 9060 . . . . 5 ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card)
86, 7mpancom 668 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ dom card)
9 eqid 2771 . . . . 5 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
109dmmptss 5775 . . . 4 dom (𝑥𝐵𝐶) ⊆ 𝐵
11 ssnum 9062 . . . 4 ((𝐵 ∈ dom card ∧ dom (𝑥𝐵𝐶) ⊆ 𝐵) → dom (𝑥𝐵𝐶) ∈ dom card)
128, 10, 11sylancl 574 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ∈ dom card)
13 funmpt 6069 . . . 4 Fun (𝑥𝐵𝐶)
14 funforn 6263 . . . 4 (Fun (𝑥𝐵𝐶) ↔ (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶))
1513, 14mpbi 220 . . 3 (𝑥𝐵𝐶):dom (𝑥𝐵𝐶)–onto→ran (𝑥𝐵𝐶)
16 fodomnum 9080 . . 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 5298 . . . 4 (𝐵 ≼ ω → 𝐵 ∈ V)
19 ssdomg 8155 . . . 4 (𝐵 ∈ V → (dom (𝑥𝐵𝐶) ⊆ 𝐵 → dom (𝑥𝐵𝐶) ≼ 𝐵))
2018, 10, 19mpisyl 21 . . 3 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ 𝐵)
21 domtr 8162 . . 3 ((dom (𝑥𝐵𝐶) ≼ 𝐵𝐵 ≼ ω) → dom (𝑥𝐵𝐶) ≼ ω)
2220, 21mpancom 668 . 2 (𝐵 ≼ ω → dom (𝑥𝐵𝐶) ≼ ω)
23 domtr 8162 . 2 ((ran (𝑥𝐵𝐶) ≼ dom (𝑥𝐵𝐶) ∧ dom (𝑥𝐵𝐶) ≼ ω) → ran (𝑥𝐵𝐶) ≼ ω)
2417, 22, 23syl2anc 573 1 (𝐵 ≼ ω → ran (𝑥𝐵𝐶) ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2145  Vcvv 3351  wss 3723   class class class wbr 4786  cmpt 4863  dom cdm 5249  ran crn 5250  Ord word 5865  Oncon0 5866  Fun wfun 6025  ontowfo 6029  ωcom 7212  cdom 8107  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-card 8965  df-acn 8968
This theorem is referenced by:  1stcrest  21477  2ndcrest  21478  lly1stc  21520  abrexct  29834  ldgenpisyslem1  30566  saliuncl  41059  meadjiun  41200
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