Theorem entrfil 9114

Description: Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8948). (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 8898	. 2 ⊢ (𝐵 ≈ 𝐶 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶)
2		bren 8898	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)
3		exdistrv 1957	. . . . 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 1959	. . . . . 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 6799	. . . . . . . . 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 9108	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑓 ∘ 𝑔):𝐴–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
8	6, 7	sylan2 594	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
9	8	exlimivv 1934	. . . . . 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 596	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (∃𝑔 𝑔:𝐴–1-1-onto→𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶)) → 𝐴 ≈ 𝐶)
12	11	3impb 1115	. . 3 ⊢ ((𝐴 ∈ Fin ∧ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
13	2, 12	syl3an2b 1407	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
14	1, 13	syl3an3b 1408	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∃wex 1781   ∈ wcel 2114   class class class wbr 5086   ∘ ccom 5630  –1-1-onto→wf1o 6493   ≈ cen 8885  Fincfn 8888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-om 7813  df-1o 8400  df-en 8889  df-fin 8892

This theorem is referenced by:  enfii  9115  entrfi  9119  phplem2  9134