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Mirrors > Home > MPE Home > Th. List > f1dmex | Structured version Visualization version GIF version |
Description: If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 5154. (Contributed by NM, 4-Sep-2004.) |
Ref | Expression |
---|---|
f1dmex | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6549 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | frnd 6494 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵) |
3 | ssexg 5191 | . . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V) | |
4 | 2, 3 | sylan 583 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V) |
5 | 4 | ex 416 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → ran 𝐹 ∈ V)) |
6 | f1cnv 6613 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) | |
7 | f1ofo 6597 | . . . 4 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹:ran 𝐹–onto→𝐴) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–onto→𝐴) |
9 | fornex 7639 | . . 3 ⊢ (ran 𝐹 ∈ V → (◡𝐹:ran 𝐹–onto→𝐴 → 𝐴 ∈ V)) | |
10 | 5, 8, 9 | syl6ci 71 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → 𝐴 ∈ V)) |
11 | 10 | imp 410 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ◡ccnv 5518 ran crn 5520 –1-1→wf1 6321 –onto→wfo 6322 –1-1-onto→wf1o 6323 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 |
This theorem is referenced by: f1ovv 7641 f1domg 8512 ordtypelem10 8975 oiexg 8983 inf3lem7 9081 pwfseqlem4 10073 pwfseqlem5 10074 grothomex 10240 gsumzf1o 19025 dprdf1o 19147 f1lindf 20511 tsmsf1o 22750 diophrw 39700 |
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