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Theorem sbthfi 9239
Description: Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 9133). (Contributed by BTernaryTau, 4-Nov-2024.)
Assertion
Ref Expression
sbthfi ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵)

Proof of Theorem sbthfi
Dummy variables 𝑤 𝑧 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 8991 . . . . 5 Rel ≼
21brrelex1i 5741 . . . 4 (𝐴𝐵𝐴 ∈ V)
31brrelex1i 5741 . . . 4 (𝐵𝐴𝐵 ∈ V)
4 breq1 5146 . . . . . . 7 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
5 breq2 5147 . . . . . . 7 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
64, 53anbi23d 1441 . . . . . 6 (𝑧 = 𝐴 → ((𝑤 ∈ Fin ∧ 𝑧𝑤𝑤𝑧) ↔ (𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴)))
7 breq1 5146 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
86, 7imbi12d 344 . . . . 5 (𝑧 = 𝐴 → (((𝑤 ∈ Fin ∧ 𝑧𝑤𝑤𝑧) → 𝑧𝑤) ↔ ((𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴) → 𝐴𝑤)))
9 eleq1 2829 . . . . . . 7 (𝑤 = 𝐵 → (𝑤 ∈ Fin ↔ 𝐵 ∈ Fin))
10 breq2 5147 . . . . . . 7 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
11 breq1 5146 . . . . . . 7 (𝑤 = 𝐵 → (𝑤𝐴𝐵𝐴))
129, 10, 113anbi123d 1438 . . . . . 6 (𝑤 = 𝐵 → ((𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴) ↔ (𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴)))
13 breq2 5147 . . . . . 6 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
1412, 13imbi12d 344 . . . . 5 (𝑤 = 𝐵 → (((𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴) → 𝐴𝑤) ↔ ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵)))
15 vex 3484 . . . . . 6 𝑧 ∈ V
16 sseq1 4009 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
17 imaeq2 6074 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑓𝑦) = (𝑓𝑥))
1817difeq2d 4126 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑤 ∖ (𝑓𝑦)) = (𝑤 ∖ (𝑓𝑥)))
1918imaeq2d 6078 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓𝑥))))
20 difeq2 4120 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑧𝑦) = (𝑧𝑥))
2119, 20sseq12d 4017 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦) ↔ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥)))
2216, 21anbi12d 632 . . . . . . 7 (𝑦 = 𝑥 → ((𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦)) ↔ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))))
2322cbvabv 2812 . . . . . 6 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))} = {𝑥 ∣ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))}
24 eqid 2737 . . . . . 6 ((𝑓 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))}) ∪ (𝑔 ↾ (𝑧 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))}))) = ((𝑓 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))}) ∪ (𝑔 ↾ (𝑧 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))})))
25 vex 3484 . . . . . 6 𝑤 ∈ V
2615, 23, 24, 25sbthfilem 9238 . . . . 5 ((𝑤 ∈ Fin ∧ 𝑧𝑤𝑤𝑧) → 𝑧𝑤)
278, 14, 26vtocl2g 3574 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵))
282, 3, 27syl2an 596 . . 3 ((𝐴𝐵𝐵𝐴) → ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵))
29283adant1 1131 . 2 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵))
3029pm2.43i 52 1 ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  cdif 3948  cun 3949  wss 3951   cuni 4907   class class class wbr 5143  ccnv 5684  cres 5687  cima 5688  cen 8982  cdom 8983  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989
This theorem is referenced by:  domnsymfi  9240  php  9247
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