Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hfun Structured version   Visualization version   GIF version

Theorem hfun 31455
Description: The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
Assertion
Ref Expression
hfun ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴𝐵) ∈ Hf )

Proof of Theorem hfun
StepHypRef Expression
1 rankung 31443 . . 3 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
2 elhf2g 31453 . . . . 5 (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
32ibi 255 . . . 4 (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω)
4 elhf2g 31453 . . . . 5 (𝐵 ∈ Hf → (𝐵 ∈ Hf ↔ (rank‘𝐵) ∈ ω))
54ibi 255 . . . 4 (𝐵 ∈ Hf → (rank‘𝐵) ∈ ω)
6 eleq1a 2683 . . . . . 6 ((rank‘𝐵) ∈ ω → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
76adantl 481 . . . . 5 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
8 uncom 3719 . . . . . . . . . 10 ((rank‘𝐵) ∪ (rank‘𝐴)) = ((rank‘𝐴) ∪ (rank‘𝐵))
98eqeq1i 2615 . . . . . . . . 9 (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
109biimpi 205 . . . . . . . 8 (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐴))
1110eleq1d 2672 . . . . . . 7 (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω ↔ (rank‘𝐴) ∈ ω))
1211biimprcd 239 . . . . . 6 ((rank‘𝐴) ∈ ω → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
1312adantr 480 . . . . 5 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω))
14 nnord 6965 . . . . . . 7 ((rank‘𝐴) ∈ ω → Ord (rank‘𝐴))
15 nnord 6965 . . . . . . 7 ((rank‘𝐵) ∈ ω → Ord (rank‘𝐵))
16 ordtri2or2 5740 . . . . . . 7 ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
1714, 15, 16syl2an 493 . . . . . 6 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)))
18 ssequn1 3745 . . . . . . 7 ((rank‘𝐴) ⊆ (rank‘𝐵) ↔ ((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵))
19 ssequn1 3745 . . . . . . 7 ((rank‘𝐵) ⊆ (rank‘𝐴) ↔ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴))
2018, 19orbi12i 542 . . . . . 6 (((rank‘𝐴) ⊆ (rank‘𝐵) ∨ (rank‘𝐵) ⊆ (rank‘𝐴)) ↔ (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) ∨ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴)))
2117, 20sylib 207 . . . . 5 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → (((rank‘𝐴) ∪ (rank‘𝐵)) = (rank‘𝐵) ∨ ((rank‘𝐵) ∪ (rank‘𝐴)) = (rank‘𝐴)))
227, 13, 21mpjaod 395 . . . 4 (((rank‘𝐴) ∈ ω ∧ (rank‘𝐵) ∈ ω) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
233, 5, 22syl2an 493 . . 3 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ ω)
241, 23eqeltrd 2688 . 2 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (rank‘(𝐴𝐵)) ∈ ω)
25 unexg 6857 . . 3 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴𝐵) ∈ V)
26 elhf2g 31453 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ Hf ↔ (rank‘(𝐴𝐵)) ∈ ω))
2725, 26syl 17 . 2 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ((𝐴𝐵) ∈ Hf ↔ (rank‘(𝐴𝐵)) ∈ ω))
2824, 27mpbird 246 1 ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴𝐵) ∈ Hf )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cun 3538  wss 3540  Ord word 5639  cfv 5804  ωcom 6957  rankcrnk 8509   Hf chf 31449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-r1 8510  df-rank 8511  df-hf 31450
This theorem is referenced by:  hfadj  31457
  Copyright terms: Public domain W3C validator