Theorem entrfil 9126

Description: Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8954). (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 8905	. 2 ⊢ (𝐵 ≈ 𝐶 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶)
2		bren 8905	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)
3		exdistrv 1955	. . . . 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 1957	. . . . . 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 6805	. . . . . . . . 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 9120	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑓 ∘ 𝑔):𝐴–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
8	6, 7	sylan2 593	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
9	8	exlimivv 1932	. . . . . 6 ⊢ (∃𝑔∃𝑓(𝐴 ∈ Fin ∧ (𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
10	4, 9	sylbir 235	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∃𝑔∃𝑓(𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
11	3, 10	sylan2br 595	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (∃𝑔 𝑔:𝐴–1-1-onto→𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
12	11	3impb 1114	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
13	2, 12	syl3an2b 1406	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
14	1, 13	syl3an3b 1407	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∃wex 1779   ∈ wcel 2109   class class class wbr 5102   ∘ ccom 5635  –1-1-onto→wf1o 6498   ≈ cen 8892  Fincfn 8895

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-en 8896  df-fin 8899

This theorem is referenced by:  enfii  9127  entrfi  9131  phplem2  9146