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Theorem sbthfi 9237
Description: Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 9132). (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 8990 . . . . 5 Rel ≼
21brrelex1i 5745 . . . 4 (𝐴𝐵𝐴 ∈ V)
31brrelex1i 5745 . . . 4 (𝐵𝐴𝐵 ∈ V)
4 breq1 5151 . . . . . . 7 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
5 breq2 5152 . . . . . . 7 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
64, 53anbi23d 1438 . . . . . 6 (𝑧 = 𝐴 → ((𝑤 ∈ Fin ∧ 𝑧𝑤𝑤𝑧) ↔ (𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴)))
7 breq1 5151 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
86, 7imbi12d 344 . . . . 5 (𝑧 = 𝐴 → (((𝑤 ∈ Fin ∧ 𝑧𝑤𝑤𝑧) → 𝑧𝑤) ↔ ((𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴) → 𝐴𝑤)))
9 eleq1 2827 . . . . . . 7 (𝑤 = 𝐵 → (𝑤 ∈ Fin ↔ 𝐵 ∈ Fin))
10 breq2 5152 . . . . . . 7 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
11 breq1 5151 . . . . . . 7 (𝑤 = 𝐵 → (𝑤𝐴𝐵𝐴))
129, 10, 113anbi123d 1435 . . . . . 6 (𝑤 = 𝐵 → ((𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴) ↔ (𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴)))
13 breq2 5152 . . . . . 6 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
1412, 13imbi12d 344 . . . . 5 (𝑤 = 𝐵 → (((𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴) → 𝐴𝑤) ↔ ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵)))
15 vex 3482 . . . . . 6 𝑧 ∈ V
16 sseq1 4021 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
17 imaeq2 6076 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑓𝑦) = (𝑓𝑥))
1817difeq2d 4136 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑤 ∖ (𝑓𝑦)) = (𝑤 ∖ (𝑓𝑥)))
1918imaeq2d 6080 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓𝑥))))
20 difeq2 4130 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑧𝑦) = (𝑧𝑥))
2119, 20sseq12d 4029 . . . . . . . 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 2810 . . . . . 6 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))} = {𝑥 ∣ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))}
24 eqid 2735 . . . . . 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 3482 . . . . . 6 𝑤 ∈ V
2615, 23, 24, 25sbthfilem 9236 . . . . 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 1129 . 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 1086   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  cdif 3960  cun 3961  wss 3963   cuni 4912   class class class wbr 5148  ccnv 5688  cres 5691  cima 5692  cen 8981  cdom 8982  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-dom 8986  df-fin 8988
This theorem is referenced by:  domnsymfi  9238  php  9245
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