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Theorem unfilem1 8169
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 7023 . . . . . . . . . 10 ((𝑥𝐵𝐵 ∈ ω) → 𝑥 ∈ ω)
31, 2mpan2 706 . . . . . . . . 9 (𝑥𝐵𝑥 ∈ ω)
4 unfilem1.1 . . . . . . . . . 10 𝐴 ∈ ω
5 nnaord 7645 . . . . . . . . . 10 ((𝑥 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
61, 4, 5mp3an23 1413 . . . . . . . . 9 (𝑥 ∈ ω → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
73, 6syl 17 . . . . . . . 8 (𝑥𝐵 → (𝑥𝐵 ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
87ibi 256 . . . . . . 7 (𝑥𝐵 → (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵))
9 nnaword1 7655 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
10 nnord 7021 . . . . . . . . . . 11 (𝐴 ∈ ω → Ord 𝐴)
114, 10ax-mp 5 . . . . . . . . . 10 Ord 𝐴
12 nnacl 7637 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
13 nnord 7021 . . . . . . . . . . 11 ((𝐴 +𝑜 𝑥) ∈ ω → Ord (𝐴 +𝑜 𝑥))
1412, 13syl 17 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → Ord (𝐴 +𝑜 𝑥))
15 ordtri1 5718 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
1611, 14, 15sylancr 694 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
179, 16mpbid 222 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
184, 3, 17sylancr 694 . . . . . . 7 (𝑥𝐵 → ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)
198, 18jca 554 . . . . . 6 (𝑥𝐵 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
20 eleq1 2692 . . . . . . . 8 (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ↔ (𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵)))
21 eleq1 2692 . . . . . . . . 9 (𝑦 = (𝐴 +𝑜 𝑥) → (𝑦𝐴 ↔ (𝐴 +𝑜 𝑥) ∈ 𝐴))
2221notbid 308 . . . . . . . 8 (𝑦 = (𝐴 +𝑜 𝑥) → (¬ 𝑦𝐴 ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴))
2320, 22anbi12d 746 . . . . . . 7 (𝑦 = (𝐴 +𝑜 𝑥) → ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴) ↔ ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴)))
2423biimparc 504 . . . . . 6 ((((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ∧ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐴) ∧ 𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
2519, 24sylan 488 . . . . 5 ((𝑥𝐵𝑦 = (𝐴 +𝑜 𝑥)) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
2625rexlimiva 3026 . . . 4 (∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥) → (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
274, 1nnacli 7640 . . . . . . . 8 (𝐴 +𝑜 𝐵) ∈ ω
28 elnn 7023 . . . . . . . 8 ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ (𝐴 +𝑜 𝐵) ∈ ω) → 𝑦 ∈ ω)
2927, 28mpan2 706 . . . . . . 7 (𝑦 ∈ (𝐴 +𝑜 𝐵) → 𝑦 ∈ ω)
30 nnord 7021 . . . . . . . . 9 (𝑦 ∈ ω → Ord 𝑦)
31 ordtri1 5718 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝑦) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
3210, 30, 31syl2an 494 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ¬ 𝑦𝐴))
33 nnawordex 7663 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴𝑦 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
3432, 33bitr3d 270 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
354, 29, 34sylancr 694 . . . . . 6 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦𝐴 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦))
36 eleq1 2692 . . . . . . . . . 10 ((𝐴 +𝑜 𝑥) = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐴 +𝑜 𝐵) ↔ 𝑦 ∈ (𝐴 +𝑜 𝐵)))
376, 36sylan9bb 735 . . . . . . . . 9 ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥𝐵𝑦 ∈ (𝐴 +𝑜 𝐵)))
3837biimprcd 240 . . . . . . . 8 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑥𝐵))
39 eqcom 2633 . . . . . . . . . . 11 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
4039biimpi 206 . . . . . . . . . 10 ((𝐴 +𝑜 𝑥) = 𝑦𝑦 = (𝐴 +𝑜 𝑥))
4140adantl 482 . . . . . . . . 9 ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑦 = (𝐴 +𝑜 𝑥))
4241a1i 11 . . . . . . . 8 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → 𝑦 = (𝐴 +𝑜 𝑥)))
4338, 42jcad 555 . . . . . . 7 (𝑦 ∈ (𝐴 +𝑜 𝐵) → ((𝑥 ∈ ω ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝑥𝐵𝑦 = (𝐴 +𝑜 𝑥))))
4443reximdv2 3013 . . . . . 6 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝑦 → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
4535, 44sylbid 230 . . . . 5 (𝑦 ∈ (𝐴 +𝑜 𝐵) → (¬ 𝑦𝐴 → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥)))
4645imp 445 . . . 4 ((𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴) → ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥))
4726, 46impbii 199 . . 3 (∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
48 unfilem1.3 . . . 4 𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
49 ovex 6633 . . . 4 (𝐴 +𝑜 𝑥) ∈ V
5048, 49elrnmpti 5340 . . 3 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐵 𝑦 = (𝐴 +𝑜 𝑥))
51 eldif 3570 . . 3 (𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴) ↔ (𝑦 ∈ (𝐴 +𝑜 𝐵) ∧ ¬ 𝑦𝐴))
5247, 50, 513bitr4i 292 . 2 (𝑦 ∈ ran 𝐹𝑦 ∈ ((𝐴 +𝑜 𝐵) ∖ 𝐴))
5352eqriv 2623 1 ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wrex 2913  cdif 3557  wss 3560  cmpt 4678  ran crn 5080  Ord word 5684  (class class class)co 6605  ωcom 7013   +𝑜 coa 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-oadd 7510
This theorem is referenced by:  unfilem2  8170
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