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Theorem unfilem1 9261
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 7818 . . . . . . . . . 10 ((𝑥𝐵𝐵 ∈ ω) → 𝑥 ∈ ω)
31, 2mpan2 690 . . . . . . . . 9 (𝑥𝐵𝑥 ∈ ω)
4 unfilem1.1 . . . . . . . . . 10 𝐴 ∈ ω
5 nnaord 8571 . . . . . . . . . 10 ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥𝐵 ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
61, 4, 5mp3an23 1454 . . . . . . . . 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 8581 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
10 nnord 7815 . . . . . . . . . 10 (𝐴 ∈ ω → Ord 𝐴)
11 nnacl 8563 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω)
12 nnord 7815 . . . . . . . . . . 11 ((𝐴 +o 𝑥) ∈ ω → Ord (𝐴 +o 𝑥))
1311, 12syl 17 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +o 𝑥))
14 ordtri1 6355 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord (𝐴 +o 𝑥)) → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
1510, 13, 14syl2an2r 684 . . . . . . . . 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 588 . . . . . . 7 (𝑥𝐵 → ¬ (𝐴 +o 𝑥) ∈ 𝐴)
188, 17jca 513 . . . . . 6 (𝑥𝐵 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴))
19 eleq1 2826 . . . . . . . 8 (𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ↔ (𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵)))
20 eleq1 2826 . . . . . . . . 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 481 . . . . . 6 ((((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ∧ ¬ (𝐴 +o 𝑥) ∈ 𝐴) ∧ 𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
2418, 23sylan 581 . . . . 5 ((𝑥𝐵𝑦 = (𝐴 +o 𝑥)) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
2524rexlimiva 3145 . . . 4 (∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥) → (𝑦 ∈ (𝐴 +o 𝐵) ∧ ¬ 𝑦𝐴))
264, 1nnacli 8566 . . . . . . . 8 (𝐴 +o 𝐵) ∈ ω
27 elnn 7818 . . . . . . . 8 ((𝑦 ∈ (𝐴 +o 𝐵) ∧ (𝐴 +o 𝐵) ∈ ω) → 𝑦 ∈ ω)
2826, 27mpan2 690 . . . . . . 7 (𝑦 ∈ (𝐴 +o 𝐵) → 𝑦 ∈ ω)
29 nnord 7815 . . . . . . . . 9 (𝑦 ∈ ω → Ord 𝑦)
30 ordtri1 6355 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
3110, 29, 30syl2an 597 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
32 nnawordex 8589 . . . . . . . 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 588 . . . . . 6 (𝑦 ∈ (𝐴 +o 𝐵) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝑦))
35 eleq1 2826 . . . . . . . . . 10 ((𝐴 +o 𝑥) = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐴 +o 𝐵) ↔ 𝑦 ∈ (𝐴 +o 𝐵)))
366, 35sylan9bb 511 . . . . . . . . 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 215 . . . . . . . . 9 ((𝐴 +o 𝑥) = 𝑦𝑦 = (𝐴 +o 𝑥))
4039adantl 483 . . . . . . . 8 ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → 𝑦 = (𝐴 +o 𝑥))
4137, 40jca2 515 . . . . . . 7 (𝑦 ∈ (𝐴 +o 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +o 𝑥) = 𝑦) → (𝑥𝐵𝑦 = (𝐴 +o 𝑥))))
4241reximdv2 3162 . . . . . 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 408 . . . 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 7395 . . . 4 (𝐴 +o 𝑥) ∈ V
4846, 47elrnmpti 5920 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐵 𝑦 = (𝐴 +o 𝑥))
49 eldif 3925 . . 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 205  wa 397   = wceq 1542  wcel 2107  wrex 3074  cdif 3912  wss 3915  cmpt 5193  ran crn 5639  Ord word 6321  (class class class)co 7362  ωcom 7807   +o coa 8414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-oadd 8421
This theorem is referenced by:  unfilem2  9262
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