Theorem entrfil 8931

Description: Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8747). (Contributed by BTernaryTau, 10-Sep-2024.)

Assertion

Ref	Expression
entrfil	⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)

Proof of Theorem entrfil

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8701	. 2 ⊢ (𝐵 ≈ 𝐶 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶)
2		bren 8701	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)
3		exdistrv 1960	. . . . 5 ⊢ (∃𝑔∃𝑓(𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶) ↔ (∃𝑔 𝑔:𝐴–1-1-onto→𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶))
4		19.42vv 1962	. . . . . 6 ⊢ (∃𝑔∃𝑓(𝐴 ∈ Fin ∧ (𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶)) ↔ (𝐴 ∈ Fin ∧ ∃𝑔∃𝑓(𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶)))
5		f1oco 6722	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐶 ∧ 𝑔:𝐴–1-1-onto→𝐵) → (𝑓 ∘ 𝑔):𝐴–1-1-onto→𝐶)
6	5	ancoms 458	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶) → (𝑓 ∘ 𝑔):𝐴–1-1-onto→𝐶)
7		f1oenfi 8926	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑓 ∘ 𝑔):𝐴–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
8	6, 7	sylan2 592	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
9	8	exlimivv 1936	. . . . . 6 ⊢ (∃𝑔∃𝑓(𝐴 ∈ Fin ∧ (𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
10	4, 9	sylbir 234	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∃𝑔∃𝑓(𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
11	3, 10	sylan2br 594	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (∃𝑔 𝑔:𝐴–1-1-onto→𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
12	11	3impb 1113	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
13	2, 12	syl3an2b 1402	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
14	1, 13	syl3an3b 1403	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085  ∃wex 1783   ∈ wcel 2108   class class class wbr 5070   ∘ ccom 5584  –1-1-onto→wf1o 6417   ≈ cen 8688  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-en 8692  df-fin 8695

This theorem is referenced by:  enfii  8932  entrfi  8936