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Theorem unfilem1 9343
Description: Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
unfilem1.1 𝐴 ∈ ω
unfilem1.2 𝐵 ∈ ω
unfilem1.3 𝐹 = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
Assertion
Ref Expression
unfilem1 ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem unfilem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 unfilem1.2 . . . . . . . . . 10 𝐵 ∈ ω
2 elnn 7898 . . . . . . . . . 10 ((𝑥𝐵𝐵 ∈ ω) → 𝑥 ∈ ω)
31, 2mpan2 691 . . . . . . . . 9 (𝑥𝐵𝑥 ∈ ω)
4 unfilem1.1 . . . . . . . . . 10 𝐴 ∈ ω
5 nnaord 8657 . . . . . . . . . 10 ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
61, 4, 5mp3an23 1455 . . . . . . . . 9 (𝑥 ∈ ω → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
73, 6syl 17 . . . . . . . 8 (𝑥𝐵 → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
87ibi 267 . . . . . . 7 (𝑥𝐵 → (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))
9 nnaword1 8667 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
10 nnord 7895 . . . . . . . . . 10 (𝐴 ∈ ω → Ord 𝐴)
11 nnacl 8649 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
12 nnord 7895 . . . . . . . . . . 11 ((𝐴 +o 𝑥) ∈ ω → Ord (𝐴 +o 𝑥))
1311, 12syl 17 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +o 𝑥))
14 ordtri1 6417 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord (𝐴 +o 𝑥)) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
1510, 13, 14syl2an2r 685 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
169, 15mpbid 232 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
174, 3, 16sylancr 587 . . . . . . 7 (𝑥𝐵 → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
188, 17jca 511 . . . . . 6 (𝑥𝐵 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
19 eleq1 2829 . . . . . . . 8 (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
20 eleq1 2829 . . . . . . . . 9 (𝑦 = (𝐴 +o 𝑥) → (𝑦𝐴 ↔ (𝐴 +o 𝑥) ∈ 𝐴))
2120notbid 318 . . . . . . . 8 (𝑦 = (𝐴 +o 𝑥) → (¬ 𝑦𝐴 ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
2219, 21anbi12d 632 . . . . . . 7 (𝑦 = (𝐴 +o 𝑥) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴) ↔ ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴)))
2322biimparc 479 . . . . . 6 ((((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴) ∧ 𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
2418, 23sylan 580 . . . . 5 ((𝑥𝐵𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
2524rexlimiva 3147 . . . 4 (∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
264, 1nnacli 8652 . . . . . . . 8 (𝐴 +o 𝐵) ∈ ω
27 elnn 7898 . . . . . . . 8 ((𝑦 ∈ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝑦 ∈ ω)
2826, 27mpan2 691 . . . . . . 7 (𝑦 ∈ (𝐴 +o 𝐵) → 𝑦 ∈ ω)
29 nnord 7895 . . . . . . . . 9 (𝑦 ∈ ω → Ord 𝑦)
30 ordtri1 6417 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
3110, 29, 30syl2an 596 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
32 nnawordex 8675 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
3331, 32bitr3d 281 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
344, 28, 33sylancr 587 . . . . . 6 (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
35 eleq1 2829 . . . . . . . . . 10 ((𝐴 +o 𝑥) = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ↔ 𝑦 ∈ (𝐴 +o 𝐵)))
366, 35sylan9bb 509 . . . . . . . . 9 ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥𝐵𝑦 ∈ (𝐴 +o 𝐵)))
3736biimprcd 250 . . . . . . . 8 (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑥𝐵))
38 eqcom 2744 . . . . . . . . . 10 ((𝐴 +o 𝑥) = 𝑦𝑦 = (𝐴 +o 𝑥))
3938biimpi 216 . . . . . . . . 9 ((𝐴 +o 𝑥) = 𝑦𝑦 = (𝐴 +o 𝑥))
4039adantl 481 . . . . . . . 8 ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑦 = (𝐴 +o 𝑥))
4137, 40jca2 513 . . . . . . 7 (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥𝐵𝑦 = (𝐴 +o 𝑥))))
4241reximdv2 3164 . . . . . 6 (𝑦 ∈ (𝐴 +o 𝐵) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦 → ∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥)))
4334, 42sylbid 240 . . . . 5 (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦𝐴 → ∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥)))
4443imp 406 . . . 4 ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴) → ∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥))
4525, 44impbii 209 . . 3 (∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
46 unfilem1.3 . . . 4 𝐹 = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
47 ovex 7464 . . . 4 (𝐴 +o 𝑥) ∈ V
4846, 47elrnmpti 5973 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥))
49 eldif 3961 . . 3 (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
5045, 48, 493bitr4i 303 . 2 (𝑦 ∈ ran 𝐹𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴))
5150eqriv 2734 1 ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  cdif 3948  wss 3951  cmpt 5225  ran crn 5686  Ord word 6383  (class class class)co 7431  ωcom 7887   +o coa 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510
This theorem is referenced by:  unfilem2  9344
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