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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 5289. (Contributed by NM, 4-Sep-2004.) |
Ref | Expression |
---|---|
f1dmex | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6798 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | frnd 6735 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵) |
3 | ssexg 5327 | . . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V) | |
4 | 2, 3 | sylan 578 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → ran 𝐹 ∈ V) |
5 | 4 | ex 411 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐵 ∈ 𝐶 → ran 𝐹 ∈ V)) |
6 | f1cnv 6868 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) | |
7 | f1ofo 6851 | . . . 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 | focdmex 7965 | . . 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 405 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 ◡ccnv 5681 ran crn 5683 –1-1→wf1 6550 –onto→wfo 6551 –1-1-onto→wf1o 6552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 |
This theorem is referenced by: f1ovv 7967 f1domg 8999 ordtypelem10 9558 oiexg 9566 inf3lem7 9665 pwfseqlem4 10693 pwfseqlem5 10694 grothomex 10860 gsumzf1o 19874 dprdf1o 19996 f1lindf 21763 tsmsf1o 24069 diophrw 42210 |
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