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Theorem unfilem1 9313
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 7869 . . . . . . . . . 10 ((𝑥𝐵𝐵 ∈ ω) → 𝑥 ∈ ω)
31, 2mpan2 688 . . . . . . . . 9 (𝑥𝐵𝑥 ∈ ω)
4 unfilem1.1 . . . . . . . . . 10 𝐴 ∈ ω
5 nnaord 8622 . . . . . . . . . 10 ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
61, 4, 5mp3an23 1452 . . . . . . . . 9 (𝑥 ∈ ω → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
73, 6syl 17 . . . . . . . 8 (𝑥𝐵 → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
87ibi 266 . . . . . . 7 (𝑥𝐵 → (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵))
9 nnaword1 8632 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
10 nnord 7866 . . . . . . . . . 10 (𝐴 ∈ ω → Ord 𝐴)
11 nnacl 8614 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
12 nnord 7866 . . . . . . . . . . 11 ((𝐴 +o 𝑥) ∈ ω → Ord (𝐴 +o 𝑥))
1311, 12syl 17 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +o 𝑥))
14 ordtri1 6398 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord (𝐴 +o 𝑥)) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
1510, 13, 14syl2an2r 682 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
169, 15mpbid 231 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
174, 3, 16sylancr 586 . . . . . . 7 (𝑥𝐵 → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
188, 17jca 511 . . . . . 6 (𝑥𝐵 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
19 eleq1 2820 . . . . . . . 8 (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
20 eleq1 2820 . . . . . . . . 9 (𝑦 = (𝐴 +o 𝑥) → (𝑦𝐴 ↔ (𝐴 +o 𝑥) ∈ 𝐴))
2120notbid 317 . . . . . . . 8 (𝑦 = (𝐴 +o 𝑥) → (¬ 𝑦𝐴 ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
2219, 21anbi12d 630 . . . . . . 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 579 . . . . 5 ((𝑥𝐵𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
2524rexlimiva 3146 . . . 4 (∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
264, 1nnacli 8617 . . . . . . . 8 (𝐴 +o 𝐵) ∈ ω
27 elnn 7869 . . . . . . . 8 ((𝑦 ∈ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝑦 ∈ ω)
2826, 27mpan2 688 . . . . . . 7 (𝑦 ∈ (𝐴 +o 𝐵) → 𝑦 ∈ ω)
29 nnord 7866 . . . . . . . . 9 (𝑦 ∈ ω → Ord 𝑦)
30 ordtri1 6398 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
3110, 29, 30syl2an 595 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
32 nnawordex 8640 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
3331, 32bitr3d 280 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
344, 28, 33sylancr 586 . . . . . 6 (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
35 eleq1 2820 . . . . . . . . . 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 249 . . . . . . . 8 (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑥𝐵))
38 eqcom 2738 . . . . . . . . . 10 ((𝐴 +o 𝑥) = 𝑦𝑦 = (𝐴 +o 𝑥))
3938biimpi 215 . . . . . . . . 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 3163 . . . . . 6 (𝑦 ∈ (𝐴 +o 𝐵) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦 → ∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥)))
4334, 42sylbid 239 . . . . 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 208 . . 3 (∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
46 unfilem1.3 . . . 4 𝐹 = (𝑥𝐵 ↦ (𝐴 +o 𝑥))
47 ovex 7445 . . . 4 (𝐴 +o 𝑥) ∈ V
4846, 47elrnmpti 5960 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥))
49 eldif 3959 . . 3 (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
5045, 48, 493bitr4i 302 . 2 (𝑦 ∈ ran 𝐹𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴))
5150eqriv 2728 1 ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395   = wceq 1540  wcel 2105  wrex 3069  cdif 3946  wss 3949  cmpt 5232  ran crn 5678  Ord word 6364  (class class class)co 7412  ωcom 7858   +o coa 8466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-oadd 8473
This theorem is referenced by:  unfilem2  9314
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