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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 4699. (Contributed by NM, 4-Sep-2004.) |
Ref | Expression |
---|---|
f1dmex | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6014 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | frn 5966 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵) |
4 | ssexg 4732 | . . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V) | |
5 | 3, 4 | sylan 487 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V) |
6 | 5 | ex 449 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → ran 𝐹 ∈ V)) |
7 | f1cnv 6073 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) | |
8 | f1ofo 6057 | . . . 4 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹:ran 𝐹–onto→𝐴) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–onto→𝐴) |
10 | fornex 7028 | . . 3 ⊢ (ran 𝐹 ∈ V → (◡𝐹:ran 𝐹–onto→𝐴 → 𝐴 ∈ V)) | |
11 | 6, 9, 10 | syl6ci 69 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → 𝐴 ∈ V)) |
12 | 11 | imp 444 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ◡ccnv 5037 ran crn 5039 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: f1ovv 7030 f1domg 7861 ordtypelem10 8315 oiexg 8323 inf3lem7 8414 pwfseqlem4 9363 pwfseqlem5 9364 grothomex 9530 gsumzf1o 18136 dprdf1o 18254 f1lindf 19980 tsmsf1o 21758 diophrw 36340 |
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