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| Description: Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 9047). (Contributed by BTernaryTau, 10-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| entrfil | ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bren 8996 | . 2 ⊢ (𝐵 ≈ 𝐶 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) | |
| 2 | bren 8996 | . . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵) | |
| 3 | exdistrv 1954 | . . . . 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 1956 | . . . . . 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 6870 | . . . . . . . . 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 9220 | . . . . . . . 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 1931 | . . . . . 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 1405 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ ∃𝑓 𝑓:𝐵–1-1-onto→𝐶) → 𝐴 ≈ 𝐶) | 
| 14 | 1, 13 | syl3an3b 1406 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∃wex 1778 ∈ wcel 2107 class class class wbr 5142 ∘ ccom 5688 –1-1-onto→wf1o 6559 ≈ cen 8983 Fincfn 8986 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-om 7889 df-1o 8507 df-en 8987 df-fin 8990 | 
| This theorem is referenced by: enfii 9227 entrfi 9231 phplem2 9246 | 
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