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Mirrors > Home > ILE Home > Th. List > f1oen2g | GIF version |
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6173 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
f1oen2g | ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐹:A–1-1-onto→B) → A ≈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1of 5069 | . . . 4 ⊢ (𝐹:A–1-1-onto→B → 𝐹:A⟶B) | |
2 | fex2 5002 | . . . 4 ⊢ ((𝐹:A⟶B ∧ A ∈ 𝑉 ∧ B ∈ 𝑊) → 𝐹 ∈ V) | |
3 | 1, 2 | syl3an1 1167 | . . 3 ⊢ ((𝐹:A–1-1-onto→B ∧ A ∈ 𝑉 ∧ B ∈ 𝑊) → 𝐹 ∈ V) |
4 | 3 | 3coml 1110 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐹:A–1-1-onto→B) → 𝐹 ∈ V) |
5 | simp3 905 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐹:A–1-1-onto→B) → 𝐹:A–1-1-onto→B) | |
6 | f1oen3g 6170 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹:A–1-1-onto→B) → A ≈ B) | |
7 | 4, 5, 6 | syl2anc 391 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐹:A–1-1-onto→B) → A ≈ B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 884 ∈ wcel 1390 Vcvv 2551 class class class wbr 3755 ⟶wf 4841 –1-1-onto→wf1o 4844 ≈ cen 6155 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-en 6158 |
This theorem is referenced by: f1oeng 6173 enrefg 6180 en2d 6184 en3d 6185 ener 6195 f1imaen2g 6209 cnven 6224 xpcomen 6237 nnenom 8891 |
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