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Mirrors > Home > ILE Home > Th. List > f1dmex | GIF version |
Description: If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.) |
Ref | Expression |
---|---|
f1dmex | ⊢ ((𝐹:A–1-1→B ∧ B ∈ 𝐶) → A ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 5035 | . . . . . 6 ⊢ (𝐹:A–1-1→B → 𝐹:A⟶B) | |
2 | frn 4995 | . . . . . 6 ⊢ (𝐹:A⟶B → ran 𝐹 ⊆ B) | |
3 | 1, 2 | syl 14 | . . . . 5 ⊢ (𝐹:A–1-1→B → ran 𝐹 ⊆ B) |
4 | ssexg 3887 | . . . . 5 ⊢ ((ran 𝐹 ⊆ B ∧ B ∈ 𝐶) → ran 𝐹 ∈ V) | |
5 | 3, 4 | sylan 267 | . . . 4 ⊢ ((𝐹:A–1-1→B ∧ B ∈ 𝐶) → ran 𝐹 ∈ V) |
6 | 5 | ex 108 | . . 3 ⊢ (𝐹:A–1-1→B → (B ∈ 𝐶 → ran 𝐹 ∈ V)) |
7 | f1cnv 5093 | . . . . 5 ⊢ (𝐹:A–1-1→B → ◡𝐹:ran 𝐹–1-1-onto→A) | |
8 | f1ofo 5076 | . . . . 5 ⊢ (◡𝐹:ran 𝐹–1-1-onto→A → ◡𝐹:ran 𝐹–onto→A) | |
9 | 7, 8 | syl 14 | . . . 4 ⊢ (𝐹:A–1-1→B → ◡𝐹:ran 𝐹–onto→A) |
10 | fornex 5684 | . . . 4 ⊢ (ran 𝐹 ∈ V → (◡𝐹:ran 𝐹–onto→A → A ∈ V)) | |
11 | 9, 10 | syl5com 26 | . . 3 ⊢ (𝐹:A–1-1→B → (ran 𝐹 ∈ V → A ∈ V)) |
12 | 6, 11 | syld 40 | . 2 ⊢ (𝐹:A–1-1→B → (B ∈ 𝐶 → A ∈ V)) |
13 | 12 | imp 115 | 1 ⊢ ((𝐹:A–1-1→B ∧ B ∈ 𝐶) → A ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 ◡ccnv 4287 ran crn 4289 ⟶wf 4841 –1-1→wf1 4842 –onto→wfo 4843 –1-1-onto→wf1o 4844 |
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-coll 3863 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-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 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-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 |
This theorem is referenced by: f1domg 6174 |
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