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Theorem unfilem1 9254
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 7813 . . . . . . . . . 10 ((𝑥𝐵𝐵 ∈ ω) → 𝑥 ∈ ω)
31, 2mpan2 689 . . . . . . . . 9 (𝑥𝐵𝑥 ∈ ω)
4 unfilem1.1 . . . . . . . . . 10 𝐴 ∈ ω
5 nnaord 8566 . . . . . . . . . 10 ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
61, 4, 5mp3an23 1453 . . . . . . . . 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 8576 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
10 nnord 7810 . . . . . . . . . 10 (𝐴 ∈ ω → Ord 𝐴)
11 nnacl 8558 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
12 nnord 7810 . . . . . . . . . . 11 ((𝐴 +o 𝑥) ∈ ω → Ord (𝐴 +o 𝑥))
1311, 12syl 17 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +o 𝑥))
14 ordtri1 6350 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord (𝐴 +o 𝑥)) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
1510, 13, 14syl2an2r 683 . . . . . . . . 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 587 . . . . . . 7 (𝑥𝐵 → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
188, 17jca 512 . . . . . 6 (𝑥𝐵 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
19 eleq1 2825 . . . . . . . 8 (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
20 eleq1 2825 . . . . . . . . 9 (𝑦 = (𝐴 +o 𝑥) → (𝑦𝐴 ↔ (𝐴 +o 𝑥) ∈ 𝐴))
2120notbid 317 . . . . . . . 8 (𝑦 = (𝐴 +o 𝑥) → (¬ 𝑦𝐴 ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
2219, 21anbi12d 631 . . . . . . 7 (𝑦 = (𝐴 +o 𝑥) → ((𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴) ↔ ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴)))
2322biimparc 480 . . . . . 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 3144 . . . 4 (∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
264, 1nnacli 8561 . . . . . . . 8 (𝐴 +o 𝐵) ∈ ω
27 elnn 7813 . . . . . . . 8 ((𝑦 ∈ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝑦 ∈ ω)
2826, 27mpan2 689 . . . . . . 7 (𝑦 ∈ (𝐴 +o 𝐵) → 𝑦 ∈ ω)
29 nnord 7810 . . . . . . . . 9 (𝑦 ∈ ω → Ord 𝑦)
30 ordtri1 6350 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
3110, 29, 30syl2an 596 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
32 nnawordex 8584 . . . . . . . 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 587 . . . . . 6 (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
35 eleq1 2825 . . . . . . . . . 10 ((𝐴 +o 𝑥) = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ↔ 𝑦 ∈ (𝐴 +o 𝐵)))
366, 35sylan9bb 510 . . . . . . . . 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 2743 . . . . . . . . . 10 ((𝐴 +o 𝑥) = 𝑦𝑦 = (𝐴 +o 𝑥))
3938biimpi 215 . . . . . . . . 9 ((𝐴 +o 𝑥) = 𝑦𝑦 = (𝐴 +o 𝑥))
4039adantl 482 . . . . . . . 8 ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑦 = (𝐴 +o 𝑥))
4137, 40jca2 514 . . . . . . 7 (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥𝐵𝑦 = (𝐴 +o 𝑥))))
4241reximdv2 3161 . . . . . 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 407 . . . 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 7390 . . . 4 (𝐴 +o 𝑥) ∈ V
4846, 47elrnmpti 5915 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥))
49 eldif 3920 . . 3 (𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
5045, 48, 493bitr4i 302 . 2 (𝑦 ∈ ran 𝐹𝑦 ∈ ((𝐴 +o 𝐵) ∖ 𝐴))
5150eqriv 2733 1 ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3073  cdif 3907  wss 3910  cmpt 5188  ran crn 5634  Ord word 6316  (class class class)co 7357  ωcom 7802   +o coa 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-oadd 8416
This theorem is referenced by:  unfilem2  9255
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