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Theorem sbthfi 9149
Description: Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 9040). (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 8892 . . . . 5 Rel ≼
21brrelex1i 5689 . . . 4 (𝐴𝐵𝐴 ∈ V)
31brrelex1i 5689 . . . 4 (𝐵𝐴𝐵 ∈ V)
4 breq1 5109 . . . . . . 7 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
5 breq2 5110 . . . . . . 7 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
64, 53anbi23d 1440 . . . . . 6 (𝑧 = 𝐴 → ((𝑤 ∈ Fin ∧ 𝑧𝑤𝑤𝑧) ↔ (𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴)))
7 breq1 5109 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑤𝐴𝑤))
86, 7imbi12d 345 . . . . 5 (𝑧 = 𝐴 → (((𝑤 ∈ Fin ∧ 𝑧𝑤𝑤𝑧) → 𝑧𝑤) ↔ ((𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴) → 𝐴𝑤)))
9 eleq1 2822 . . . . . . 7 (𝑤 = 𝐵 → (𝑤 ∈ Fin ↔ 𝐵 ∈ Fin))
10 breq2 5110 . . . . . . 7 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
11 breq1 5109 . . . . . . 7 (𝑤 = 𝐵 → (𝑤𝐴𝐵𝐴))
129, 10, 113anbi123d 1437 . . . . . 6 (𝑤 = 𝐵 → ((𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴) ↔ (𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴)))
13 breq2 5110 . . . . . 6 (𝑤 = 𝐵 → (𝐴𝑤𝐴𝐵))
1412, 13imbi12d 345 . . . . 5 (𝑤 = 𝐵 → (((𝑤 ∈ Fin ∧ 𝐴𝑤𝑤𝐴) → 𝐴𝑤) ↔ ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵)))
15 vex 3448 . . . . . 6 𝑧 ∈ V
16 sseq1 3970 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝑧𝑥𝑧))
17 imaeq2 6010 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑓𝑦) = (𝑓𝑥))
1817difeq2d 4083 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑤 ∖ (𝑓𝑦)) = (𝑤 ∖ (𝑓𝑥)))
1918imaeq2d 6014 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑔 “ (𝑤 ∖ (𝑓𝑦))) = (𝑔 “ (𝑤 ∖ (𝑓𝑥))))
20 difeq2 4077 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑧𝑦) = (𝑧𝑥))
2119, 20sseq12d 3978 . . . . . . . 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 2806 . . . . . 6 {𝑦 ∣ (𝑦𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑦))) ⊆ (𝑧𝑦))} = {𝑥 ∣ (𝑥𝑧 ∧ (𝑔 “ (𝑤 ∖ (𝑓𝑥))) ⊆ (𝑧𝑥))}
24 eqid 2733 . . . . . 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 3448 . . . . . 6 𝑤 ∈ V
2615, 23, 24, 25sbthfilem 9148 . . . . 5 ((𝑤 ∈ Fin ∧ 𝑧𝑤𝑤𝑧) → 𝑧𝑤)
278, 14, 26vtocl2g 3530 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵))
282, 3, 27syl2an 597 . . 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 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3444  cdif 3908  cun 3909  wss 3911   cuni 4866   class class class wbr 5106  ccnv 5633  cres 5636  cima 5637  cen 8883  cdom 8884  Fincfn 8886
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 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-en 8887  df-dom 8888  df-fin 8890
This theorem is referenced by:  domnsymfi  9150  php  9157
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