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Theorem sucdom2OLD 9123
Description: Obsolete version of sucdom2 9244 as of 4-Dec-2024. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sucdom2OLD (𝐴𝐵 → suc 𝐴𝐵)

Proof of Theorem sucdom2OLD
Dummy variables 𝑤 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sdomdom 9021 . . 3 (𝐴𝐵𝐴𝐵)
2 brdomi 9000 . . 3 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
31, 2syl 17 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
4 relsdom 8993 . . . . . . 7 Rel ≺
54brrelex1i 5740 . . . . . 6 (𝐴𝐵𝐴 ∈ V)
65adantr 480 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐴 ∈ V)
7 vex 3483 . . . . . . 7 𝑓 ∈ V
87rnex 7933 . . . . . 6 ran 𝑓 ∈ V
98a1i 11 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ran 𝑓 ∈ V)
10 f1f1orn 6858 . . . . . . 7 (𝑓:𝐴1-1𝐵𝑓:𝐴1-1-onto→ran 𝑓)
1110adantl 481 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝑓:𝐴1-1-onto→ran 𝑓)
12 f1of1 6846 . . . . . 6 (𝑓:𝐴1-1-onto→ran 𝑓𝑓:𝐴1-1→ran 𝑓)
1311, 12syl 17 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝑓:𝐴1-1→ran 𝑓)
14 f1dom2g 9011 . . . . 5 ((𝐴 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓)
156, 9, 13, 14syl3anc 1372 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐴 ≼ ran 𝑓)
16 sdomnen 9022 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐴𝐵)
1716adantr 480 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ¬ 𝐴𝐵)
18 ssdif0 4365 . . . . . . . 8 (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅)
19 simplr 768 . . . . . . . . . . 11 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴1-1𝐵)
20 f1f 6803 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
2120frnd 6743 . . . . . . . . . . . . 13 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
2219, 21syl 17 . . . . . . . . . . . 12 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓𝐵)
23 simpr 484 . . . . . . . . . . . 12 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓)
2422, 23eqssd 4000 . . . . . . . . . . 11 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵)
25 dff1o5 6856 . . . . . . . . . . 11 (𝑓:𝐴1-1-onto𝐵 ↔ (𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵))
2619, 24, 25sylanbrc 583 . . . . . . . . . 10 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴1-1-onto𝐵)
27 f1oen3g 9008 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴1-1-onto𝐵) → 𝐴𝐵)
287, 26, 27sylancr 587 . . . . . . . . 9 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴𝐵)
2928ex 412 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐵 ⊆ ran 𝑓𝐴𝐵))
3018, 29biimtrrid 243 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴𝐵))
3117, 30mtod 198 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅)
32 neq0 4351 . . . . . 6 (¬ (𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
3331, 32sylib 218 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
34 snssi 4807 . . . . . . 7 (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓))
35 vex 3483 . . . . . . . . 9 𝑤 ∈ V
36 en2sn 9082 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤})
376, 35, 36sylancl 586 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → {𝐴} ≈ {𝑤})
384brrelex2i 5741 . . . . . . . . . 10 (𝐴𝐵𝐵 ∈ V)
3938adantr 480 . . . . . . . . 9 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐵 ∈ V)
40 difexg 5328 . . . . . . . . 9 (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
41 ssdomg 9041 . . . . . . . . 9 ((𝐵 ∖ ran 𝑓) ∈ V → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
4239, 40, 413syl 18 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
43 endomtr 9053 . . . . . . . 8 (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
4437, 42, 43syl6an 684 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
4534, 44syl5 34 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
4645exlimdv 1932 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
4733, 46mpd 15 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
48 disjdif 4471 . . . . 5 (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅
4948a1i 11 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅)
50 undom 9100 . . . 4 (((𝐴 ≼ ran 𝑓 ∧ {𝐴} ≼ (𝐵 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
5115, 47, 49, 50syl21anc 837 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
52 df-suc 6389 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5352a1i 11 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴}))
54 undif2 4476 . . . 4 (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓𝐵)
5521adantl 481 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
56 ssequn1 4185 . . . . 5 (ran 𝑓𝐵 ↔ (ran 𝑓𝐵) = 𝐵)
5755, 56sylib 218 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (ran 𝑓𝐵) = 𝐵)
5854, 57eqtr2id 2789 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
5951, 53, 583brtr4d 5174 . 2 ((𝐴𝐵𝑓:𝐴1-1𝐵) → suc 𝐴𝐵)
603, 59exlimddv 1934 1 (𝐴𝐵 → suc 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wex 1778  wcel 2107  Vcvv 3479  cdif 3947  cun 3948  cin 3949  wss 3950  c0 4332  {csn 4625   class class class wbr 5142  ran crn 5685  suc csuc 6385  1-1wf1 6557  1-1-ontowf1o 6559  cen 8983  cdom 8984  csdm 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-suc 6389  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-en 8987  df-dom 8988  df-sdom 8989
This theorem is referenced by: (None)
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